\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 76, pp. 1--32.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/76\hfil Variational methods for a resonant problem] {Variational methods for a resonant problem\\ with the $p$-Laplacian in $\mathbb{R}^N$} \author[B.\ Alziary, J.\ Fleckinger, P.\ Tak\'a\v{c}\hfil EJDE-2004/76\hfilneg] {B\'en\'edicte Alziary, Jacqueline Fleckinger, Peter Tak\'a\v{c}} % in alphabetical order \address{B\'en\'edicte Alziary \hfill\break CEREMATH \& UMR MIP\\ Universit\'e Toulouse~1 -- Sciences Sociales\\ 21 all\'ees de Brienne, F--31042 Toulouse Cedex, France} \email{alziary@univ-tlse1.fr} \address{Jacqueline Fleckinger \hfill\break CEREMATH \& UMR MIP \\ Universit\'e Toulouse~1 -- Sciences Sociales\\ 21 all\'ees de Brienne, F--31042 Toulouse Cedex, France} \email{jfleck@univ-tlse1.fr} \address{Peter Tak\'a\v{c} \hfill\break Fachbereich Mathematik\\ Universit\"at Rostock\\ Universit\"atsplatz 1, D--18055 Rostock, Germany} \email{peter.takac@mathematik.uni-rostock.de} \date{} \thanks{Submitted March 19, 2004. Published May 26, 2004.} \subjclass[2000]{35P30, 35J20, 47J10, 47J30} \keywords{$p$-Laplacian, degenerate quasilinear Cauchy problem, \hfill\break\indent Fredholm alternative, $(p-1)$-homogeneous problem at resonance, saddle point geometry, \hfill\break\indent improved Poincar\'e inequality, second-order Taylor formula} \begin{abstract} The solvability of the resonant Cauchy problem $$ - \Delta_p u = \lambda_1 m(|x|) |u|^{p-2} u + f(x) \quad\hbox{in } \mathbb{R}^N ;\quad u\in D^{1,p}(\mathbb{R}^N), $$ in the entire Euclidean space $\mathbb{R}^N$ ($N\geq 1$) is investigated as a part of the Fredholm alternative at the first (smallest) eigenvalue $\lambda_1$ of the positive $p$-Laplacian $-\Delta_p$ on $D^{1,p}(\mathbb{R}^N)$ relative to the weight $m(|x|)$. Here, $\Delta_p$ stands for the $p$-Laplacian, $m\colon \mathbb{R}_+\to \mathbb{R}_+$ is a weight function assumed to be radially symmetric, $m\not\equiv 0$ in $\mathbb{R}_+$, and $f\colon \mathbb{R}^N\to \mathbb{R}$ is a given function satisfying a suitable integrability condition. The weight $m(r)$ is assumed to be bounded and to decay fast enough as $r\to +\infty$. Let $\varphi_1$ denote the (positive) eigenfunction associated with the (simple) eigenvalue $\lambda_1$ of $-\Delta_p$. If $\int_{\mathbb{R}^N} f\varphi_1 \,{\rm d}x =0$, we show that problem has at least one solution $u$ in the completion $D^{1,p}(\mathbb{R}^N)$ of $C_{\mathrm{c}}^1(\mathbb{R}^N)$ endowed with the norm $(\int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x)^{1/p}$. To establish this existence result, we employ a saddle point method if $1 < p < 2$, and an improved Poincar\'e inequality if $2\leq p< N$. We use weighted Lebesgue and Sobolev spaces with weights depending on $\varphi_1$. The asymptotic behavior of $\varphi_1(x)= \varphi_1(|x|)$ as $|x|\to \infty$ plays a crucial role. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \numberwithin{equation}{section} \newcommand{\eqdef}{\stackrel{{\mathrm {def}}}{=}} \section{Introduction} \label{s:Intro} Spectral problems involving quasilinear degenerate or singular elliptic operators have been an interesting subject of investigation for quite some time; see e.g.\ {\sc Dr\'abek}~\cite{Drabek-1} or {\sc Fu\v{c}\'{\i}k} et al.~\cite{FucikNSS}. In our present work we focus our attention on the solvability of the Cauchy problem % \begin{equation} - \Delta_p u = \lambda\, m(x)\, |u|^{p-2} u + f(x) \;\mbox{ in } \mathbb{R}^N ;\qquad u\in D^{1,p}(\mathbb{R}^N) , \label{e:BVP.l} \end{equation} % in the entire Euclidean space $\mathbb{R}^N$ ($N\geq 1$). Here, $\Delta_p$ stands for the $p$-Laplacian defined by $\Delta_p u\equiv \mathop{\rm div} ( |\nabla u|^{p-2} \nabla u )$, $1
0$, the Sobolev imbedding $D^{1,p}(\mathbb{R}^N) \hookrightarrow L^p(\mathbb{R}^N; m)$ turns out to be compact, where $L^p(\mathbb{R}^N; m)$ denotes the weighted Lebesgue space of all measurable functions $u\colon \mathbb{R}^N\to \mathbb{R}$ with the norm % \[ \| u\|_{ L^{p}(\mathbb{R}^N; m) } \eqdef \Big( \int_{\mathbb{R}^N} |u(x)|^p\, m(x) \,{\rm d}x \Big)^{1/p} < \infty . \] % Hence, the Rayleigh quotient % \begin{equation} \lambda_1\eqdef \inf \Big\{ \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x \colon u\in D^{1,p}(\mathbb{R}^N) \;\mbox{ with }\; \int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x = 1 \Big\} \label{def.lam_1} \end{equation} % is positive and gives the first (smallest) eigenvalue $\lambda_1$ of $-\Delta_p$ relative to the weight $m$. Now take $f$ from the dual space $D^{-1,p'}(\mathbb{R}^N)$ of $D^{1,p}(\mathbb{R}^N)$, $p'= p/(p-1)$, with respect to the standard duality $\langle \,\cdot\, ,\,\cdot\, \rangle$ induced by the inner product on $L^2(\mathbb{R}^N)$. If $-\infty < \lambda < \lambda_1$ then the energy functional corresponding to equation \eqref{e:BVP.l}, % \begin{equation} \mathcal{J}_{\lambda}(u) \eqdef \frac{1}{p} \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x - \frac{\lambda}{p} \int_{\mathbb{R}^N} |u|^p\, m(x) \,{\rm d}x - \int_{\mathbb{R}^N} f(x) u \,{\rm d}x \label{def.jl} \end{equation} % defined for $u\in D^{1,p}(\mathbb{R}^N)$, is weakly lower semicontinuous and coercive on $D^{1,p}(\mathbb{R}^N)$. Thus, $\mathcal{J}_{\lambda}$ possesses a global minimizer which provides a weak solution to equation \eqref{e:BVP.l}. The critical case $\lambda = \lambda_1$ is much more complicated when $p\not= 2$ because the linear Fredholm alternative cannot be applied. First, one has to have sufficient information on the first eigenvalue $\lambda_1$; we refer the reader to {\sc Fleckinger} et al.~\cite[Sect.\ 2 and~3]{FMST} or {\sc Stavrakakis} and {\sc de~Th\'elin} \cite{StavrThelin}. One has % \begin{equation} - \Delta_p \varphi_1 = \lambda_1\, m(x)\, |\varphi_1|^{p-2} \varphi_1 \;\mbox{ in } \mathbb{R}^N ;\qquad \varphi_1\in D^{1,p}(\mathbb{R}^N)\setminus \{ 0\} , \label{e:varphi.l_1} \end{equation} % and the eigenvalue $\lambda_1$ is simple, by a result due to {\sc Anane} \cite[Th\'eor\`eme~1, p.~727]{Anane-1} and later generalized by {\sc Lindqvist} \cite[Theorem 1.3, p.~157]{Lindqvist}. Moreover, the corresponding eigenfunction $\varphi_1$ can be normalized by $\|\varphi_1\|_{ L^p(\mathbb{R}^N; m) } = 1$ and $\varphi_1 > 0$ in $\mathbb{R}^N$, owing to the strong maximum principle \cite[Prop.\ 3.2.1 and 3.2.2, p.~801]{Tolksdorf-1} or \cite[Theorem~5, p.~200]{Vazquez}. We decompose the unknown function $u\in D^{1,p}(\mathbb{R}^N)$ as a direct sum % \begin{equation} \label{ortho:u} \begin{gathered} u = u^\parallel\cdot \varphi_1 + u^\top\quad \mbox{ where }\\ u^\parallel = \int_{\mathbb{R}^N} u\, \varphi_1\, \mu(x) \,{\rm d}x \in \mathbb{R} \;\mbox{ and }\; \int_{\mathbb{R}^N} u^\top\, \varphi_1\, \mu(x) \,{\rm d}x = 0 , \end{gathered} \end{equation} % with the weight $\mu(x)$ given by $\mu\eqdef \varphi_1^{p-2}\, m$. It is quite natural that we treat the two components, $u^\parallel$ and $u^\top$, differently. The linearization of the equation % \begin{equation} - \Delta_p u = \lambda_1\, m(x)\, |u|^{p-2} u + f(x) \;\mbox{ in } \mathbb{R}^N ;\qquad u\in D^{1,p}(\mathbb{R}^N) , \label{e:BVP.l_1} \end{equation} % about $u^\parallel\cdot \varphi_1$, and the corresponding ``quadratization'' of the functional $\mathcal{J}_{\lambda_1}$, play an important role in our approach. We will also see that the orthogonality condition % \begin{equation} \int_{\mathbb{R}^N} f\, \varphi_1\, \mu \,{\rm d}x \equiv \int_{\mathbb{R}^N} f\, \varphi_1^{p-1}\, m \,{\rm d}x = 0 \label{e:f.phi_1=0} \end{equation} % for $f$ and $\varphi_1$ relative to the measure $\mu(x) \,{\rm d}x$ is sufficient, but not necessary for the solvability of problem~\eqref{e:BVP.l_1}. Similarly as in {\sc Dr\'abek} and {\sc Holubov\'a} \cite{DrabHolub} for $1
2} for $2\leq p < N$ and Theorem~\ref{thm-Exist:p<2} for $1 < p < 2\leq N$, and some properties of the energy functional $\mathcal{J}_{\lambda}$ needed to establish the solvability, as well. Naturally, our approach requires the compactness of several Sobolev imbeddings in $\mathbb{R}^N$ with weights (Proposition \ref{prop-Compact}) which we prove in Section~\ref{s:pf-Compact}. In Section~\ref{s:Auxiliary} we establish a few auxiliary results for the quadratization of $\mathcal{J}_{\lambda_1}$. We use this quadratization to verify the improved Poincar\'e inequality (Lemma~\ref{lem-Poincare}) for $2\leq p < N$ in Section~\ref{s:Impr_Poinc}. From this inequality we derive Theorem~\ref{thm-Exist:p>2} in Section~\ref{s:pr-Exist:p>2}. For $1
0} \makeatother There exist constants $\delta > 0$ and $C>0$ such that % \begin{equation} 0 < m(r)\leq \frac{C}{ (1+r)^{p+\delta} } \quad\mbox{for almost all } 0\leq r < \infty . \label{ineq:m>0} \end{equation} \end{enumerate} \begin{remark}\label{rem-hyp:m>0}\begingroup\rm In fact, in hypothesis \eqref{hyp:m>0} above, instead of $m(r) > 0$ for almost all $0\leq r < \infty$, it suffices to assume only $m\geq 0$ a.e.\ in $\mathbb{R}^N$ and $m$ does not vanish identically near zero, i.e., for every $r_0 > 0$ we have $m\not\equiv 0$ in $(0,r_0)$. However, if $m\equiv 0$ on a set $S\subset \mathbb{R}_+$ of positive Lebesgue measure, then the weighted spaces $\mathcal{H}_{\varphi_1} = L^2(\mathbb{R}^N; \varphi_1^{p-2} m)$, $L^p(\mathbb{R}^N; m)$, etc.\ defined below become linear spaces with a seminorm only. Moreover, all functions from their dual spaces $ \mathcal{H}_{\varphi_1}' = L^{2}\big( \mathbb{R}^N; \varphi_1^{2-p} m^{-1} \big)$, $L^{p'}( \mathbb{R}^N; m^{-1/(p-1)} )$, etc., respectively, must vanish identically (i.e., almost everywhere) in the ``spherical shell'' $\{ x\in \mathbb{R}^N\colon |x|\in S\}$. This would make our presentation much less clear; therefore, we have decided to leave the necessary amendments in our arguments to an interested reader. \endgroup \end{remark} \subsection{The first eigenfunction $\varphi_1$} \label{ss:phi_1} Under hypothesis \eqref{hyp:m>0}, the first eigenvalue $\lambda_1$ of $-\Delta_p$ on $\mathbb{R}^N$ relative to the weight $m(|x|)$ is simple and the eigenfunction $\varphi_1$ associated with $\lambda_1$ is commonly called a ``ground state'' for the Cauchy problem~\eqref{e:varphi.l_1}. The simplicity of $\lambda_1$ forces $\varphi_1(x) = \varphi_1(|x|)$ radially symmetric in $\mathbb{R}^N$. Hence, the eigenvalue problem \eqref{e:varphi.l_1} is equivalent to % \begin{equation*} \begin{split} {}- ( |\varphi_1'|^{p-2} \varphi_1' )' - \frac{N-1}{r}\, |\varphi_1'|^{p-2} \varphi_1' = \lambda_1\, m(r)\, \varphi_1^{p-1} \quad\mbox{for } r>0 ; \\ \mbox{subject to}\quad \int_0^\infty |\varphi_1'(r)|^p\, r^{N-1} \,{\rm d}r < \infty \quad\mbox{and }\quad \varphi_1(r) \to 0 \mbox{ as } r\to \infty . \end{split} \end{equation*} % It can be further rewritten as % \begin{equation} \label{ev:phi_1.rad} \begin{split} {}- ( r^{N-1}\, |\varphi_1'|^{p-2} \varphi_1' )' = \lambda_1\, m(r)\, r^{N-1}\, \varphi_1^{p-1} \quad\mbox{for } r>0 ; \\ \varphi_1'(r)\to 0 \mbox{ as } r\to 0 \quad\mbox{and }\quad \varphi_1(r) \to 0 \mbox{ as } r\to \infty . \end{split} \end{equation} % Recalling hypothesis \eqref{hyp:m>0}, from \eqref{ev:phi_1.rad} we can deduce the following simple facts. %%%%% LEMMA - phi_1' < 0 and Simple Facts %%%% \begin{lemma}\label{lem-phi_1'} Let $1
0$. % \end{lemma} To determine the asymptotic behavior of $\varphi_1(r)$ as $r\to \infty$, we will investigate the corresponding nonlinear eigenvalue problem \eqref{ev:phi_1.rad} in Appendix~\ref{s:Asymptotic}. Higher smoothness of $\varphi_1\colon \mathbb{R}_+\to (0,\infty)$ can be obtained directly by integrating equation \eqref{ev:phi_1.rad}: $\varphi_1\in C^{1,\beta}(\mathbb{R}_+)$ with $\beta = \min\{ 1 ,\, \frac{1}{p-1} \}$. We refer to {\sc Man\'a\-se\-vich} and {\sc Tak\'a\v{c}} \cite[Eq.~(33)]{ManTakac} for details. \subsection{Notation} \label{ss:Notation} The closure and boundary of a set $S\subset \mathbb{R}^N$ are denoted by $\overline{S}$ and $\partial S$, respectively. We denote by $B_{\varrho}\eqdef \{ x\in \mathbb{R}^N\colon |x| < \varrho\}$ the ball of radius $0 < \varrho < \infty$. All Banach and Hilbert spaces used in this article are real. Given an integer $k\geq 0$ and $0\leq \alpha\leq 1$, we denote by $C^{k,\alpha}(\mathbb{R}^N)$ the linear space of all $k$-times continuously differentiable functions $u\colon \mathbb{R}^N\to \mathbb{R}$ whose all (classical) partial derivatives of order $\leq k$ are locally $\alpha$-H\"older continuous on $\mathbb{R}^N$. As usual, we abbreviate $C^k(\mathbb{R}^N) \equiv C^{k,0}(\mathbb{R}^N)$. The linear subspace of $C^k(\mathbb{R}^N)$ consisting of all $C^k$ functions $u\colon \mathbb{R}^N\to \mathbb{R}$ with compact support is denoted by $C_{\mathrm{c}}^k(\mathbb{R}^N)$. For $1
0$, such that for arbitrary vectors ${\bf a}, {\bf b}, {\bf v}\in \mathbb{R}^N$ we have % \begin{equation} \label{1-s.A.geom:p>2} \begin{split} c_p\cdot \big( \max_{0\leq s\leq 1} |{\bf a} + s {\bf b}| \big)^{p-2} |{\bf v}|^2 & \leq \int_0^1 \langle {\bf A}({\bf a} + s {\bf b}) {\bf v}, {\bf v} \rangle (1-s) \,{\rm d}s \\ & \leq \frac{p-1}{2} \big( \max_{0\leq s\leq 1} |{\bf a} + s {\bf b}| \big)^{p-2} |{\bf v}|^2 . \end{split} \end{equation} % On the other hand, given any $1 < p < 2$, there exists a constant $c_p > 0$, such that for arbitrary vectors ${\bf a}, {\bf b}, {\bf v}\in \mathbb{R}^N$, with $|{\bf a}| + |{\bf b}| > 0$, we have % \begin{equation} \label{1-s.A.geom:p<2} \begin{split} \frac{p-1}{2} \big( \max_{0\leq s\leq 1} |{\bf a} + s {\bf b}| \big)^{p-2} |{\bf v}|^2 & \leq \int_0^1 \langle {\bf A}({\bf a} + s {\bf b}) {\bf v}, {\bf v} \rangle (1-s) \,{\rm d}s \\ & \leq c_p\cdot \big( \max_{0\leq s\leq 1} |{\bf a} + s {\bf b}| \big)^{p-2} |{\bf v}|^2 . \end{split} \end{equation} % These inequalities are needed to treat the linearization of $-\Delta_p$ at $\varphi_1$ below. Next, as in \cite[Sect.~1]{Takac-1}, we rewrite the first and second terms of the energy functional $\mathcal{J}_{\lambda_1}$ using the integral forms of the first- and second\--order Taylor formulas; we set % \begin{equation} \mathcal{F}(u) \eqdef \frac{1}{p} \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x - \frac{\lambda_1}{p} \int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x ,\quad u\in D^{1,p}(\mathbb{R}^N) . \label{def.J} \end{equation} % We need to treat the Taylor formulas for $p\geq 2$ and $1
0$. Furthermore, our definition~\eqref{def.lam_1} of $\lambda_1$ and eq.~\eqref{J''.phi_1} guarantee $\mathcal{Q}_{t\phi}(\phi,\phi) \geq 0$ for all $t\in \mathbb{R}\setminus \{ 0\}$. Letting $t\to 0$ we arrive at % \begin{equation} \mathcal{Q}_0 (\phi,\phi) \geq 0 \quad\mbox{for all }\, \phi\in D^{1,p}(\mathbb{R}^N) . \label{Q.geq.0} \end{equation} % \noindent {\it Case\/} $1
2} Let $2\leq p < N$. If $f\in \mathcal{D}_{\varphi_1}'$ satisfies $\langle f, \varphi_1\rangle = 0$, then problem~\eqref{e:BVP.l_1} possesses a weak solution $u\in D^{1,p}(\mathbb{R}^N)$. % \end{theorem} This is a part of the Fredholm alternative for $-\Delta_p$ at $\lambda_1$. The proof is given in Section~\ref{s:pr-Exist:p>2}. In a bounded domain $\Omega\subset \mathbb{R}^N$, this theorem is due to {\sc Fleckinger} and {\sc Tak\'a\v{c}} \cite[Theorem 3.3, p.~958]{FleckTakac-1}. The orthogonality condition $\langle f, \varphi_1\rangle = 0$ is sufficient, but {\em not\/} necessary to obtain existence for problem~\eqref{e:BVP.l_1} provided $p\neq 2$, according to recent results obtained in {\sc Dr\'abek}, {\sc Girg} and {\sc Man\'a\-se\-vich} \cite[Theorem 1.3]{DrabGirgMan} for $N=1$, in {\sc Dr\'abek} and {\sc Holubov\'a} \cite[Theorem 1.1]{DrabHolub} for any $N\geq 1$ and $1
2} \begingroup\rm For $2\leq p < N$, the hypothesis $f\in \mathcal{D}_{\varphi_1}'$ is fulfilled, for example, if $f = f_1 + f_2$ where $f_1\in L^2(B_\varepsilon; m^{-1})$ and $f_1\equiv 0$ in $\mathbb{R}^N\setminus B_\varepsilon$, and $f_2\equiv 0$ in $B_\varepsilon$ and % \begin{math} f_2\in L^2\big( \mathbb{R}^N\setminus B_\varepsilon; r^{ - N + \frac{N-p}{p-1} } \big) \end{math} % for some $0 < \varepsilon\leq 1$. This claim follows from the imbeddings in Lemma~\ref{lem-Compact} combined with the asymptotic formulas in Proposition~\ref{prop-Asympt}, where % \begin{math} \mathcal{H}_{\varphi_1}' = L^{2}\big( \mathbb{R}^N; \varphi_1^{2-p} m^{-1} \big) \end{math} % is the dual space of $\mathcal{H}_{\varphi_1}$, and $L^{2}\left( \mathbb{R}^N; |\varphi_1'|^{-p} \varphi_1^2 \right)$ is the dual space of $L^{2}\left( \mathbb{R}^N; |\varphi_1'|^p \varphi_1^{-2} \right)$. \endgroup \end{example} %%%%% Fredholm ALTERNATIVE at $\lambda_1$ (Theorem) \begin{theorem}\label{thm-Exist:p<2} Let $N\geq 2$ and\/ $1 < p < 2$. Assume that $f^{\#}\in D^{-1,p'}(\mathbb{R}^N)$ satisfies $\langle f^{\#}, \varphi_1\rangle = 0$ and $f^{\#}\not\equiv 0$ in $\mathbb{R}^N$. Then there exist two numbers $\delta\equiv \delta(f^{\#}) > 0$ and $\varrho\equiv \varrho(f^{\#}) > 0$ such that problem \eqref{e:BVP.l} with $f = f^{\#} + \zeta\, m\varphi_1^{p-1}$ has at least one solution whenever $\lambda\in (\lambda_1 - \delta, \lambda_1 + \delta)$ and $\zeta\in (-\varrho,\varrho)$. % \end{theorem} The proof of this theorem is given in Section~\ref{s:pr-Exist:p<2}. %%%%% Fredholm ALTERNATIVE at $\lambda_1$ (Remark) \begin{remark}\label{rem-Exist:p<2} \begingroup\rm In the situation of {\rm Theorem \ref{thm-Exist:p<2}}, if $\lambda\in (\lambda_1 - \delta, \lambda_1)$ and $\zeta\in (-\varrho,\varrho)$, then problem \eqref{e:BVP.l} has {\it at least three\/} solutions $u_1, u_2, u_3\in D^{1,p}(\mathbb{R}^N)$, such that % \[ \int_{\mathbb{R}^N} u_2\, \varphi_1^{p-1}\, m \,{\rm d}x < \int_{\mathbb{R}^N} u_1\, \varphi_1^{p-1}\, m \,{\rm d}x < \int_{\mathbb{R}^N} u_3\, \varphi_1^{p-1}\, m \,{\rm d}x , \] % $u_1$ is a saddle point (which will be obtained in the proof of Theorem \ref{thm-Exist:p<2}) and $u_2, u_3$ are local minimizers for the functional $\mathcal{J}_{\lambda}$ on $D^{1,p}(\mathbb{R}^N)$. The proof of this claim is given in Section \ref{s:pr-Exist:p<2}, {\S}\ref{ss:Fredh_Mount}, after the proof of Theorem \ref{thm-Exist:p<2}. \endgroup \end{remark} %%%%% Fredholm ALTERNATIVE at $\lambda_1$ (Example) \begin{example}\label{exam-Exist:p<2} \begingroup\rm For $1 < p\leq 2$, the hypothesis $f\in \mathcal{D}_{\varphi_1}'$ is fulfilled if $|x|\, f(x)\in L^{p'}(\mathbb{R}^N)$ with $p'= p/(p-1)$, by the imbedding $D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; |x|^{-p})$ in Lemma~\ref{lem-D^p.L^p}. \endgroup \end{example} The proofs of both theorems above hinge on the following imbeddings with weights. %%%%% PROPOSITION - Compact Imbedding \begin{proposition}\label{prop-Compact} Let $1
0} be satisfied. Then the following two imbeddings are compact: % \begin{itemize} % \item[{\rm (a)}] $D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; m)$; % \item[{\rm (b)}] $\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$. \end{itemize} \end{proposition} The proof of this proposition is given in Section~\ref{s:pf-Compact}. The reader is referred to {\sc Berger} and {\sc Schechter} \cite[Proof of Theorem 2.4, p.~277]{BergSchecht}, {\sc Fleckinger}, {\sc Gossez}, and {\sc de~Th\'elin} \cite[Lemma 2.3]{FGdT}, or {\sc Schechter} \cite{Schecht-1, Schecht-2} for related imbeddings and compactness results. \subsection{Properties of the corresponding energy functional} \label{ss:J-Energy} Weak solutions in $D^{1,p}(\mathbb{R}^N)$ to the Dirichlet boundary value problem \eqref{e:BVP.l_1} with $f\in D^{-1,p'}(\mathbb{R}^N)$ correspond to critical points of the energy functional $\mathcal{J}_{\lambda_1} \colon D^{1,p}(\mathbb{R}^N)\to \mathbb{R}$ defined in \eqref{def.jl} with $\lambda = \lambda_1$. Owing to the imbeddings in Proposition~\ref{prop-Compact}, all expressions in \eqref{def.jl} are meaningful. For the cases $2\leq p < N$ and $1 < p < 2\leq N$, the geometry of the functional $\mathcal{J}_{\lambda_1}$ is completely different; cf.\ {\sc Fleckinger} and {\sc Tak\'a\v{c}} \cite[Theorem 3.1, p.~957]{FleckTakac-1} and {\sc Dr\'abek} and {\sc Holubov\'a} \cite[Theorem 1.1, p.~184]{DrabHolub}, respectively, in a bounded domain $\Omega\subset \mathbb{R}^N$. In the former case, we have the following analogue of the {\em improved Poincar\'e inequality\/} from \cite[Theorem 3.1, p.~957]{FleckTakac-1}, which is of independent interest. %%%%% An Improved Poincare INEQUALITY (Lemma) \begin{lemma}\label{lem-Poincare} Let $2\leq p < N$ and let hypothesis \eqref{hyp:m>0} be satisfied. Then there exists a constant $c\equiv c(p,m) > 0$ such that the inequality % \begin{equation} \begin{split} & \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x - \lambda_1 \int_{\mathbb{R}^N} |u|^p\, m(x) \,{\rm d}x \\ & \geq c \Big( | u^\parallel |^{p-2} \int_{\mathbb{R}^N} |\nabla\varphi_1(x)|^{p-2} |\nabla u^\top|^2 \,{\rm d}x + \int_{\mathbb{R}^N} |\nabla u^\top|^p \,{\rm d}x \Big) \end{split} \label{e:Poincare} \end{equation} % holds for all\/ $u\in D^{1,p}(\mathbb{R}^N)$. % \end{lemma} Here, a function $u\in D^{1,p}(\mathbb{R}^N)$ is decomposed as the direct sum \eqref{ortho:u}. If the constant $c$ in~\eqref{e:Poincare} is replaced by zero, one obtains the classical Poincar\'e inequality; see e.g.\ {\sc Gilbarg} and {\sc Trudinger} \cite[Ineq.\ (7.44), p.~164]{GilbargTrud}. In analogy with the case $p=2$, the {\em improved Poincar\'e inequality\/} \eqref{e:Poincare} guarantees the solvability of the Cauchy boundary value problem \eqref{e:BVP.l_1} in the special case when $f\in \mathcal{D}_{\varphi_1}'$ satisfies $\langle f, \varphi_1\rangle = 0$. On the other hand, the ``singular'' case $1 < p < 2\leq N$ is much different and has to be treated by a minimax method introduced in {\sc Tak\'a\v{c}} \cite[Sect.~7]{Takac-1}. It uses the fact that the functional $\mathcal{J}_{\lambda_1}$ still remains coercive on % \begin{equation} D^{1,p}(\mathbb{R}^N)^\top \eqdef \Big\{ u\in D^{1,p}(\mathbb{R}^N) \colon \int_{\mathbb{R}^N} u\, \varphi_1^{p-1}\, m \,{\rm d}x = 0 \Big\} , \label{def.D^top} \end{equation} % the complement of $\mathop{\rm lin} \{ \varphi_1\}$ in $D^{1,p}(\mathbb{R}^N)$ with respect to the direct sum~\eqref{ortho:u}, viz.\ % \begin{math} D^{1,p}(\mathbb{R}^N) \penalty-1000 = \mathop{\rm lin} \{ \varphi_1\} \oplus D^{1,p}(\mathbb{R}^N)^\top \end{math}. % The following notion introduced in {\sc Dr\'abek} and {\sc Holubov\'a} \cite[Def.\ 2.1, p.~185]{DrabHolub} is crucial. %%%%% Simple SADDLE Point Geometry (Definition) \begin{definition}\label{def-Saddle_Geom} \begingroup\rm We say that a continuous functional $\mathcal{E}\colon D^{1,p}(\mathbb{R}^N)\to \mathbb{R}$ has a {\em simple saddle point geometry\/} if we can find $u,v\in D^{1,p}(\mathbb{R}^N)$ such that % \begin{gather*} \int_{\mathbb{R}^N} u\, \varphi_1^{p-1}\, m \,{\rm d}x < 0 < \int_{\mathbb{R}^N} v\, \varphi_1^{p-1}\, m \,{\rm d}x \quad\mbox{and } \\ \max\{ \mathcal{E}(u) ,\, \mathcal{E}(v) \} < \inf \left\{ \mathcal{E}(w)\colon w\in D^{1,p}(\mathbb{R}^N)^\top \right\} . \end{gather*} % \endgroup \end{definition} Note that on any continuous path $\theta\colon [-1,1]\to D^{1,p}(\mathbb{R}^N)$ with $\theta(-1) = u$ and $\theta(1) = v$ there is a point $w = \theta(t_0)\in D^{1,p}(\mathbb{R}^N)^\top$ for some $t_0\in [-1,1]$. Hence, $\max\{ \mathcal{E}(u) ,\, \mathcal{E}(v) \} < \mathcal{E}(w)$ shows that the function $\mathcal{E}\circ \theta\colon [-1,1]\to \mathbb{R}$ attains its maximum at some $t'\in (-1,1)$. The following result is essential; in fact it replaces Lemma~\ref{lem-Poincare}. For a bounded domain $\Omega\subset \mathbb{R}^N$, it was shown in {\sc Dr\'abek} and {\sc Holubov\'a} \cite[Lemma 2.1, p.~185]{DrabHolub}. % %%%%% Simple SADDLE Point Geometry (Lemma) % \begin{lemma}\label{lem-Saddle_Geom} Let\/ $1 < p < 2\leq N$. Assume $f\in D^{-1,p'}(\mathbb{R}^N)$ with\/ $\langle f, \varphi_1\rangle = 0$ and\/ $f\not\equiv 0$ in~$\mathbb{R}^N$. Then the functional\/ $\mathcal{J}_{\lambda_1}$ has a simple saddle point geometry. Moreover, it is unbounded from below on $D^{1,p}(\mathbb{R}^N)$. % \end{lemma} % Its proof will be given in Section \ref{s:pr-Exist:p<2}, {\S}\ref{ss:Saddle_Geom}. For $1
0} be satisfied. Then the following imbeddings are continuous: % \begin{gather} D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; |x|^{-p}) \hookrightarrow L^{p}(\mathbb{R}^N; m) ; \label{imb:D^p.L^p-m}\\ D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p^*}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; m) , \label{imb:D^p.L^p^*} \end{gather} % where $p^* = Np / (N-p)$ denotes the critical Sobolev exponent. % \end{lemma} % % \begin{proof} The imbedding $L^{p}(\mathbb{R}^N; |x|^{-p}) \hookrightarrow L^{p}(\mathbb{R}^N; m)$ follows from inequality~\eqref{ineq:m>0}. By a classical result ({\sc Gilbarg} and {\sc Trudinger} \cite[Theorem 7.10, p.~166]{GilbargTrud}), the imbedding $D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p^*}(\mathbb{R}^N)$ is continuous. Notice that $(p/p^*) + (p/N) = 1$. Finally, given an arbitrary function $u\in C_{\mathrm{c}}^0(\mathbb{R}^N)$, we combine the H\"older inequality with \eqref{ineq:m>0} to estimate % \begin{align*} \int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x &\leq \Big( \int_{\mathbb{R}^N} |u|^{p^*} \,{\rm d}x \Big)^{ p/p^* } \Big( \int_{\mathbb{R}^N} m^{N/p} \,{\rm d}x \Big)^{ p/N } \\ & \leq C \Big( \int_{\mathbb{R}^N} |u|^{p^*} \,{\rm d}x \Big)^{ p/p^* } \Big( \int_{\mathbb{R}^N} (1 + |x|)^{ - N ( 1 + \frac{\delta}{p} ) } \,{\rm d}x \Big)^{ p/N } . \end{align*} % The continuity of the imbedding $L^{p^*}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; m)$ follows because $C_{\mathrm{c}}^0(\mathbb{R}^N)$ is dense in $L^{p^*}(\mathbb{R}^N)$. \end{proof} %%%%% LEMMA - Compact Imbedding \begin{lemma}\label{lem-H^p.L^p} Let hypothesis \eqref{hyp:m>0} be satisfied. Then we have the following imbeddings: % \begin{itemize} % \item[{\rm (i)}] $\mathcal{H}_{\varphi_1} \hookrightarrow L^p(\mathbb{R}^N; m)$ if\/ $1
2$. As above, for $u\in C_{\mathrm{c}}^0(\mathbb{R}^N)$ we estimate % \begin{equation*} \int_{\mathbb{R}^N} u^2\, \varphi_1^{p-2}\, m \,{\rm d}x \leq \Big( \int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x \Big)^{ 2/p } \Big( \int_{\mathbb{R}^N} \varphi_1^{p}\, m \,{\rm d}x \Big)^{ (p-2)/p } = \| u\|_{ L^{p}(\mathbb{R}^N; m) }^2 . \end{equation*} % The lemma is proved. \end{proof} %%%%% LEMMA - Compact Imbedding \begin{lemma}\label{lem-D_1.D^p} Let hypothesis \eqref{hyp:m>0} be satisfied. The following imbeddings hold true: % \begin{itemize} % \item[{\rm (i)}] $\mathcal{D}_{\varphi_1} \hookrightarrow D^{1,p}(\mathbb{R}^N)$ if\/ $1
2$. Given $u\in C_{\mathrm{c}}^0(\mathbb{R}^N)$, we estimate \begin{align*} \int_{\mathbb{R}^N} |\nabla u|^2\, |\varphi_1'(r)|^{p-2} \,{\rm d}x &\leq \Big( \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x \Big)^{ 2/p } \Big( \int_{\mathbb{R}^N} |\varphi_1'|^{p} \,{\rm d}x \Big)^{ (p-2)/p }\\ &= \lambda_1^{ (p-2)/p }\, \| u\|_{ \mathcal{D}_{\varphi_1} }^2 . \end{align*} % This proves the lemma. \end{proof} %%%%% LEMMA - Compact Imbedding \begin{lemma}\label{lem-Compact} Let\/ $1
0} be satisfied. Then both imbeddings $\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$ and % \begin{math} \mathcal{D}_{\varphi_1} \hookrightarrow L^{2}\left( \mathbb{R}^N; |\varphi_1'|^p \varphi_1^{-2} \right) \end{math} % are continuous. % \end{lemma} % % \begin{proof} We need to distinguish between the cases $1
0} guarantees also the compactness of both imbeddings $D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; m)$ and $\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$ for $1
0$ sufficiently large as is shown in the following lemma. %%%%% LEMMA - Cut-off: Compact Imbedding \begin{lemma}\label{lem-Cut-off} Let\/ $1
0} be satisfied. Then there exist constants $C_2 > 0$, $C_3 > 0$ and $R_1 > 0$, such that for all\/ $\varrho\geq R_1$ we have % \begin{gather} \| \psi_{\varrho} u \|_{ D^{1,p}(\mathbb{R}^N) } \leq C_2\, \| u\|_{ D^{1,p}(\mathbb{R}^N) } \quad\mbox{for all } u\in D^{1,p}(\mathbb{R}^N) ; \label{est:T_psi,p}\\ \| \psi_{\varrho} u \|_{ \mathcal{D}_{\varphi_1} } \leq C_3\, \| u\|_{ \mathcal{D}_{\varphi_1} } \quad\mbox{for all } u\in \mathcal{D}_{\varphi_1} . \label{est:T_psi} \end{gather} \end{lemma} % \begin{proof} We give the proof for the case $1
0$. For an arbitrary function $u\in D^{1,p}(\mathbb{R}^N)$ we have $$ \nabla (\psi_{\varrho} u) = \psi_{\varrho}(r)\, \nabla u(x) + u(x)\, \psi_{\varrho}'(r)\, r^{-1} x \quad\mbox{for $x\in \mathbb{R}^N$ and $r = |x|$. } $$ Therefore, by the Minkowski inequality followed by \eqref{est:psi'} and the Hardy inequality \eqref{e:Hardy}, we have % \begin{equation} \label{ineq:T_psi,p} \begin{aligned} \| \psi_{\varrho} u \|_{ D^{1,p}(\mathbb{R}^N) } &= \Big( \int_{\mathbb{R}^N} |\nabla (\psi_{\varrho} u)|^p \,{\rm d}x \Big)^{1/p}\\ & \leq \Big( \int_{\mathbb{R}^N} |\psi_{\varrho}|^p |\nabla u|^p \,{\rm d}x \Big)^{1/p} + \Big( \int_{\mathbb{R}^N} |\psi_{\varrho}'|^p |u|^p \,{\rm d}x \Big)^{1/p} \\ & \leq \| u\|_{ D^{1,p}(\mathbb{R}^N) } + C_1 \Big( \int_{\mathbb{R}^N} |u(x)|^p\, |x|^{-p} \,{\rm d}x \Big)^{1/p}\\ &\leq C_2\, \| u\|_{ D^{1,p}(\mathbb{R}^N) } , \end{aligned} \end{equation} % where $C_2 = 1 + p C_1 / (N-p)$. This proves \eqref{est:T_psi,p}. Similarly, for every $u\in \mathcal{D}_{\varphi_1}$ we have % \begin{equation} \label{ineq:T_psi} \begin{aligned} \| \psi_{\varrho} u \|_{ \mathcal{D}_{\varphi_1} } &= \big( \int_{\mathbb{R}^N} |\varphi_1'|^{p-2} |\nabla (\psi_{\varrho} u)|^2 \,{\rm d}x \big)^{1/2} \\ & \leq \Big( \int_{\mathbb{R}^N} |\varphi_1'|^{p-2} |\psi_{\varrho}|^2 |\nabla u|^2 \,{\rm d}x \Big)^{1/2} + \Big( \int_{\mathbb{R}^N} |\varphi_1'|^{p-2} |\psi_{\varrho}'|^2 u^2 \,{\rm d}x \Big)^{1/2}\\ & \leq \| u\|_{ \mathcal{D}_{\varphi_1} } + \Big( \int_{\mathbb{R}^N} |\varphi_1'|^{p-2} |\psi_{\varrho}'|^2 u^2 \,{\rm d}x \Big)^{1/2} . \end{aligned} \end{equation} % The last integral is estimated as follows. Using the limit formula \eqref{e:u'/u.infty} we have % \begin{equation} \label{e:psi'/psi.infty} \varphi_1^{-1} |\varphi_1'| \geq \frac{N-p}{ 2(p-1)r } \quad\mbox{for all } r\geq R_1 , \end{equation} % where $R_1 > 0$ is a sufficiently large constant. We combine this inequality with \eqref{est:psi'} to conclude that % \begin{equation} \label{est:psi'.phi_1} | \psi_{\varrho}'(r) | \leq C_4\, \varphi_1^{-1} |\varphi_1'| \quad\mbox{for all } r\geq R_1 , \end{equation} % where $C_4 = 2 (p-1) C_1 / (N-p)$. Applying this estimate to the last integral in \eqref{ineq:T_psi}, and recalling $\psi_{\varrho}'(r) = 0$ whenever $0\leq r\leq \varrho$, for every $\varrho\geq R_1$ we get % \begin{equation*} \| \psi_{\varrho} u \|_{ \mathcal{D}_{\varphi_1} } \leq \| u\|_{ \mathcal{D}_{\varphi_1} } + C_4 \Big( \int_{\mathbb{R}^N} |\varphi_1'|^p |\varphi_1|^{-2} u^2 \,{\rm d}x \Big)^{1/2} . \end{equation*} % Finally, we invoke inequality \eqref{ineq:Imbedd} to estimate the last integral. The desired estimate \eqref{est:T_psi} follows with the constant $C_3 > 0$ given by $C_3 = 1 + 2 C_4$. \end{proof} Denoting by $J$ ($J_{\varphi_1}$, respectively) the continuous imbedding \\ $D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; m)$ ($\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$), we now show that the operators % \begin{equation*} J T_{\varrho}\colon D^{1,p}(\mathbb{R}^N) \to L^{p}(\mathbb{R}^N; m) \qquad ( J_{\varphi_1} T_{\varrho}\colon \mathcal{D}_{\varphi_1} \to \mathcal{H}_{\varphi_1} ) \end{equation*} % converge to $J$ ($J_{\varphi_1}$) in the uniform operator topology as $\varrho\to \infty$. %%%%% LEMMA - Cut-off: Convergence \begin{lemma}\label{lem-Cut-off:Conv} Let\/ $1
0} be satisfied.
Then, as $\varrho\to \infty$, we have
%
\begin{gather}
\| (1 - \psi_{\varrho}) u \|_{ L^{p}(\mathbb{R}^N; m) } \to 0
\quad\mbox{uniformly for }\
\| u\|_{ D^{1,p}(\mathbb{R}^N) } \leq 1 ;
\label{est:I-T_psi,p} \\
\| (1 - \psi_{\varrho}) u \|_{ \mathcal{H}_{\varphi_1} } \to 0
\quad\mbox{uniformly for }\
\| u\|_{ \mathcal{D}_{\varphi_1} } \leq 1 .
\label{est:I-T_psi}
\end{gather}
%
\end{lemma}
%
%
\begin{proof}
From hypothesis \eqref{hyp:m>0} we get
%
\begin{equation*}
m(r)\, r^p
\leq \frac{C\, r^p}{ (1+r)^{p+\delta} }
< \frac{C}{ (1+r)^{\delta} }
\quad\mbox{for all } r>0 .
\end{equation*}
%
Hence, for any $\varrho > 0$,
%
\begin{align*}
\int_{ |x|\geq \varrho } |u|^p\, m \,{\rm d}x
& \leq \frac{C}{ (1 + \varrho)^{\delta} }
\int_{ |x|\geq \varrho } |u|^p\, |x|^{-p} \,{\rm d}x
\\
& \leq \frac{C}{ (1 + \varrho)^{\delta} }
\big( \frac{p}{N-p}\big)^{p}
\| u\|_{ D^{1,p}(\mathbb{R}^N) }^{p} ,
\end{align*}
%
by the Hardy inequality \eqref{e:Hardy}.
Letting $\varrho\to \infty$ we obtain
the convergence \eqref{est:I-T_psi,p}.
Similarly as above, we combine
hypothesis \eqref{hyp:m>0} and inequality \eqref{e:psi'/psi.infty}
to compare the weights
%
\begin{equation*}
\frac{ \varphi_1(r)^{p-2}\, m(r) }
{ |\varphi_1'(r)|^{p}\, \varphi_1(r)^{-2} }
\leq \frac{C_5\, r^p}{ (1+r)^{p+\delta} }
< \frac{C_5}{ (1+r)^{\delta} }
\quad\mbox{for all } r\geq R_1 ,
\end{equation*}
%
where
$$
C_5 =\big( \frac{ 2 (p-1)}{N-p}\big)^p C .
$$
We use this inequality to estimate the second integral on
the left\--hand side in \eqref{ineq:Imbedd},
thus arriving at
%
\begin{equation*}
\lambda_1\int_{\mathbb{R}^N}
u^2\, \varphi_1^{p-2}\, m \,{\rm d}x
+ \frac{ (1 + \varrho)^{\delta} }{ 2 C_5 }
\int_{ |x|\geq \varrho }
u^2\, \varphi_1^{p-2}\, m \,{\rm d}x
\leq 2\, \| u\|_{ \mathcal{D}_{\varphi_1} }^2
\end{equation*}
%
for every $\varrho\geq R_1$.
Letting $\varrho\to \infty$ we obtain
the conclusion \eqref{est:I-T_psi} immediately.
\end{proof}
%
\subsection{Rest of the proof of Proposition~\ref{prop-Compact}}
\label{ss:pf-Compact}
According to Lemmas \ref{lem-D^p.L^p} and~\ref{lem-Compact},
it remains to show that the imbeddings
$D^{1,p}(\mathbb{R}^N) \hookrightarrow L^{p}(\mathbb{R}^N; m)$
and
$\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$
are compact.
We take advantage of the well\--known approximation theorem
(see {\sc Kato} \cite[Chapt.~III, {\S}4.2, p.~158]{Kato})
which states that the set of all compact linear operators
$S\colon X\to Y$, where $X$ and $Y$ are Banach spaces,
is a Banach space.
In our setting this means that, by Lemma \ref{lem-Cut-off:Conv},
it suffices to show that the operators
%
\begin{equation*}
J T_{\varrho}\colon D^{1,p}(\mathbb{R}^N) \to L^{p}(\mathbb{R}^N; m)
\quad\mbox{and }\quad
J_{\varphi_1} T_{\varrho}\colon
\mathcal{D}_{\varphi_1} \to \mathcal{H}_{\varphi_1} ,
\end{equation*}
%
respectively, are compact for each $\varrho > 0$ large enough.
Recall
$B_r = \{ x\in \mathbb{R}^N\colon |x| 0$.
Finally, the imbedding
$W_0^{1,2}(B_{2\varrho}) \hookrightarrow L^2(B_{2\varrho})$
being compact by Rellich's theorem, and
$L^2(B_{2\varrho}) \hookrightarrow \mathcal{H}_{\varphi_1}$
being continuous by \eqref{ineq:m>0}, we conclude that
%
\begin{math}
J_{\varphi_1} T_{\varrho}\colon
\mathcal{D}_{\varphi_1} \to \mathcal{H}_{\varphi_1}
\end{math}
%
is compact as well, whenever $\varrho\geq R_1$.
\noindent {\it Case\/} $p\geq 2$.
First, taking an arbitrary function
$u\in C^1(\mathbb{R}^N)$ with compact support, we derive
inequalities \eqref{CS:Imbedd} and \eqref{ineq:Imbedd}.
In particular, inequalities in \eqref{CS:Imbedd} entail
%
\begin{equation}\label{est:Imbedd:p>2}
\begin{aligned}
\lambda_1\int_{\mathbb{R}^N}
u^2\, \varphi_1^{p-2}\, m \,{\rm d}x
&\leq
2 \Big( \int_{\mathbb{R}^N}
|\varphi_1'|^{p-2}
\big(\frac{\partial u}{\partial r}\big)^2 \,{\rm d}x
\Big)^{1/2}
\Big( \int_{\mathbb{R}^N}
u^2\, |\varphi_1'|^{p}\, \varphi_1^{-2} \,{\rm d}x
\Big)^{1/2}\\
&\leq 2\, \| u\|_{ \mathcal{D}_{\varphi_1} }
\Big( \int_{\mathbb{R}^N}
u^2\, |\varphi_1'|^{p}\, \varphi_1^{-2} \,{\rm d}x
\Big)^{1/2} .
\end{aligned}
\end{equation}
%
We need to show that, besides inequalities \eqref{ineq:Imbedd},
we have also
%
\begin{equation}
\int_{B_R} |\varphi_1'|^{p}\, \varphi_1^{-2}\, u^2 \,{\rm d}x
\leq 9\cdot \log\genfrac{(}{)}{}0{ \varphi_1(0) }{ \varphi_1(R) }
\cdot \| u\|_{ \mathcal{D}_{\varphi_1} }^2
\quad\mbox{for every } R>0 .
\label{R:Imbedd:p>2}
\end{equation}
%
To this end, fix any $x'\in \mathbb{R}^N$ with $|x'| = 1$,
and take $x = rx'$ with $0\leq r\leq R$.
We use eq.~\eqref{ev:phi_1.rad} to compute
%
\begin{align*}
& r^{N-1}\, |\varphi_1'(r)|^{p-1}\, \varphi_1(r)^{-1}\, u(rx')^2
= - \left(
r^{N-1}\, |\varphi_1'|^{p-2} \varphi_1'
\right) \varphi_1^{-1}\, u^2\\
& = - \int_0^r \frac{\partial}{\partial s}
\left[
s^{N-1}\, |\varphi_1'(s)|^{p-2} \varphi_1(s)'\,
\varphi_1(s)^{-1}\, u(sx')^2
\right]
\,{\rm d}s\\
& = \lambda_1 \int_0^r
m(s)\, s^{N-1}\, \varphi_1(s)^{p-2}\, u(sx')^2 \,{\rm d}s \\
& \quad + \int_0^r
s^{N-1}\, |\varphi_1'(s)|^{p}\, \varphi_1(s)^{-2}\,
u(sx')^2 \,{\rm d}s \\
& \quad + 2 \int_0^r
s^{N-1}\, |\varphi_1'(s)|^{p-1}\, \varphi_1(s)^{-1}\,
u(sx')\, \frac{\partial u}{\partial s}(sx') \,{\rm d}s .
\end{align*}
%
Estimating the last integral by
the Cauchy\--Schwarz inequality, we have
%
\begin{align*}
r^{N-1}\, |\varphi_1'(r)|^{p-1}\, \varphi_1(r)^{-1}\, u(rx')^2
& \leq \lambda_1 \int_0^r
m(s)\, \varphi_1(s)^{p-2}\, u(sx')^2\, s^{N-1} \,{\rm d}s
\\
&\quad + 2 \int_0^r
|\varphi_1'(s)|^{p}\, \varphi_1(s)^{-2}\, u(sx')^2\,
s^{N-1} \,{\rm d}s
\\
&\quad + \int_0^r |\varphi_1'(s)|^{p-2}\,
\left( \frac{\partial u}{\partial s}(sx') \right)^2\,
s^{N-1} \,{\rm d}s .
\end{align*}
%
Next, setting $y = sx'$,
we integrate this inequality with respect to $x'$ over the unit sphere
$S_1 = \partial B_1\subset \mathbb{R}^N$ endowed with the surface measure
$\sigma$ to get
%
\begin{equation}
\label{p-1:phi_1.rad}
\begin{aligned}
& r^{N-1}\, |\varphi_1'(r)|^{p-1}\, \varphi_1(r)^{-1}
\int_{S_1} u(rx')^2 \,{\rm d}\sigma(x')
\\
& \leq \lambda_1 \int_{B_r}
u^2\, \varphi_1^{p-2}\, m \,{\rm d}y
+ 2 \int_{B_r}
u^2\, |\varphi_1'|^{p}\, \varphi_1^{-2} \,{\rm d}y
\\
& \quad
+ \int_{B_r} |\varphi_1'|^{p-2}\,
\big(\frac{\partial u}{\partial s}\big)^2 \,{\rm d}y
\leq 8\, \| u\|_{ \mathcal{D}_{\varphi_1} }^2
+ \| u\|_{ \mathcal{D}_{\varphi_1} }^2
= 9\, \| u\|_{ \mathcal{D}_{\varphi_1} }^2 ,
\end{aligned}
\end{equation}
%
by ineq.~\eqref{ineq:Imbedd}.
Finally, upon multiplication by $-\varphi_1'/ \varphi_1$
followed by integration over $0\leq r\leq R$,
we arrive at the desired inequality \eqref{R:Imbedd:p>2}.
Again, by Lemma \ref{lem-Cut-off}, the operators
%
\begin{math}
T_{\varrho}\colon
\mathcal{D}_{\varphi_1} \to \mathcal{D}_{\varphi_1}(B_{2\varrho})
\subset \mathcal{D}_{\varphi_1}
\end{math}
%
are uniformly bounded for all $\varrho\geq R_1$.
In order to show that
%
\begin{math}
J T_{\varrho}\colon
\mathcal{D}_{\varphi_1} \to \mathcal{H}_{\varphi_1}
\end{math}
%
is compact, it suffices to verify that the imbedding
%
\begin{math}
\mathcal{D}_{\varphi_1}(B_{2\varrho}) \hookrightarrow
\mathcal{H}_{\varphi_1}
\end{math}
%
is compact.
So let $\varrho\geq R_1$ be fixed.
Consider an arbitrary bounded sequence
$\{ u_n\}_{n=1}^\infty$ in the Hilbert space
$\mathcal{D}_{\varphi_1}(B_{2\varrho})$.
Hence, there exists a weakly convergent subsequence
denoted again by $\{ u_n\}_{n=1}^\infty$, i.e.,
$u_n\rightharpoonup u$ in
$\mathcal{D}_{\varphi_1}(B_{2\varrho})$ as $n\to \infty$.
Replacing $u_n - u$ by $u_n$, we may assume
$u_n\rightharpoonup 0$ weakly in
$\mathcal{D}_{\varphi_1}(B_{2\varrho})$.
In addition, we may assume
$\| u_n\|_{ \mathcal{D}_{\varphi_1} } \leq 1$ for all $n=1,2,\dots$.
Next, we show that $u_n\to 0$ strongly in
$L^{2}\left( B_{2\varrho} ; |\varphi_1'|^p \varphi_1^{-2} \right)$.
Choose $\varepsilon > 0$.
Fix $R_0 > 0$ small enough, such that
\[
9\cdot \log\big(\frac{\varphi_1(0)}{\varphi_1(R_0)}\big)
\leq \frac{\varepsilon}{2} ,
\]
by
$\lim_{r\to 0} \varphi_1(r) = \varphi_1(0) > 0$.
Hence, inequality \eqref{R:Imbedd:p>2} entails
%
\begin{equation}
\int_{B_{R_0}}
|\varphi_1'|^{p}\, \varphi_1^{-2}\, u_n^2 \,{\rm d}x
\leq \frac{\varepsilon}{2}
\quad\mbox{for } n=1,2,\dots .
\label{R_0:Imbedd:p>2}
\end{equation}
%
Since
$\gamma_2\eqdef \inf_{ [ R_0, 2\varrho ] } |\varphi_1'|^{p-2} > 0$,
by Lemma~\ref{lem-phi_1'},
the sequence
$\{ u_n\}_{n=1}^\infty$ is bounded in the Sobolev space
$W^{1,2}( B_{2\varrho} \setminus B_{R_0} )$,
by inequalities \eqref{ineq:Imbedd}.
The imbedding
%
\begin{math}
W^{1,2}( B_{2\varrho} \setminus B_{R_0} ) \hookrightarrow
L^2 ( B_{2\varrho} \setminus B_{R_0} )
\end{math}
%
being compact by Rellich's theorem, we conclude that
$u_n\to 0$ strongly in
$L^2( B_{2\varrho} \setminus B_{R_0} )$.
Consequently, there is an integer $n_0\geq 1$ large enough,
such that
%
\begin{equation}
\int_{ B_{2\varrho} \setminus B_{R_0} }
|\varphi_1'|^{p}\, \varphi_1^{-2}\, u_n^2 \,{\rm d}x
\leq \frac{\varepsilon}{2}
\quad\mbox{for every } n\geq n_0 .
\label{r>R_0:Imbedd:p>2}
\end{equation}
%
We combine estimates
\eqref{R_0:Imbedd:p>2} and \eqref{r>R_0:Imbedd:p>2}
to obtain
%
\begin{equation*}
\int_{ B_{2\varrho} }
|\varphi_1'|^{p}\, \varphi_1^{-2}\, u_n^2 \,{\rm d}x
\leq \varepsilon
\quad\mbox{for every } n\geq n_0 .
\end{equation*}
%
This means that $u_n\to 0$ strongly in
$L^{2}\left( B_{2\varrho} ; |\varphi_1'|^p \varphi_1^{-2} \right)$.
Finally, from inequality \eqref{est:Imbedd:p>2}
we deduce $u_n\to 0$ strongly also in $\mathcal{H}_{\varphi_1}$.
Hence, the imbedding
%
\begin{math}
\mathcal{D}_{\varphi_1}(B_{2\varrho}) \hookrightarrow
\mathcal{H}_{\varphi_1}
\end{math}
%
is compact as claimed.
We have completed the proof of Proposition~\ref{prop-Compact}.
\section{Properties of the quadratization at $\varphi_1$}
\label{s:Auxiliary}
In this section we state a few analog results to those in
{\sc Tak\'a\v{c}} \cite[Sect.~4]{Takac-1}
that are employed later in the proofs of
Theorem~\ref{thm-Exist:p>2} and Lemma~\ref{lem-Poincare}.
Note that inequality~\eqref{A.ellipt} entails
%
\begin{equation}
\label{norm:A.equiv}
\min\{ 1,\,p-1\}
\| v\|_{ \mathcal{D}_{\varphi_1} }^2
\leq \int_{\mathbb{R}^N}
\langle {\bf A} (\nabla\varphi_1) \nabla v , \nabla v
\rangle_{\mathbb{R}^N}
\,{\rm d}x
\leq
\max\{ 1,\,p-1\}
\| v\|_{ \mathcal{D}_{\varphi_1} }^2
\end{equation}
%
for $v\in \mathcal{D}_{\varphi_1}$.
Several important properties of $\mathcal{D}_{\varphi_1}$
are established below.
The following result is obvious.
%%%%% Q - nonnegative (Lemma)
\begin{lemma}\label{lem-Q-posit}
We have
$\mathcal{Q}_0 (\varphi_1,\varphi_1) = 0$ and\/
$0\leq \mathcal{Q}_0 (v,v) < \infty$ for all\/
$v\in \mathcal{D}_{\varphi_1}$.
%
\end{lemma}
%
We denote by $\mathcal{A}_{\varphi_1}$
the Lax\--Milgram representation of the symmetric bilinear form
$2\cdot \mathcal{Q}_0$ on
$\mathcal{D}_{\varphi_1}\times \mathcal{D}_{\varphi_1}$
(see \cite[Chapt.~VI, Eq.\ (2.3), p.~323]{Kato}).
In our setting this means that
%
\begin{math}
\mathcal{A}_{\varphi_1} \colon
\mathcal{D}_{\varphi_1} \to \mathcal{D}_{\varphi_1}'
\end{math}
%
is a bounded linear operator such that
%
\begin{equation}
\langle \mathcal{A}_{\varphi_1} v, w\rangle
= 2\cdot \mathcal{Q}_0 (v,w)
\quad\mbox{for all } v,w\in \mathcal{D}_{\varphi_1} .
\label{def.A}
\end{equation}
%
Identifying the dual space of
$\mathcal{D}_{\varphi_1}'$ with $\mathcal{D}_{\varphi_1}$
(see {\sc Yosida} \cite[Theorem IV.8.2, p.~113]{Yosida}),
we find that $\mathcal{A}_{\varphi_1}$
is selfadjoint in the following sense:
$$
\langle \mathcal{A}_{\varphi_1} v, w\rangle
= \langle v, \mathcal{A}_{\varphi_1} w\rangle
\quad\mbox{for all } v,w\in \mathcal{D}_{\varphi_1} .
$$
Note that our definition of $\mathcal{Q}_0$ yields
$\mathcal{A}_{\varphi_1} \varphi_1 = 0$.
Since the imbedding
$\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$
is compact,
the null space of $\mathcal{A}_{\varphi_1}$ denoted by
\[
\ker(\mathcal{A}_{\varphi_1}) =
\{ v\in \mathop{\rm dom}(\mathcal{A}_{\varphi_1})\colon
\mathcal{A}_{\varphi_1} v = 0 \}
\]
is finite\--dimensional, by the Riesz\--Schauder theorem
\cite[Theorem III.6.29, p.~187]{Kato}.
Lemma~\ref{lem-Q-posit} provides
another variational formula for $\lambda_1$, namely,
%
\begin{equation}
\lambda_1 = \inf
\Big\{
\frac{ \int_{\mathbb{R}^N}
\langle {\bf A}(\nabla\varphi_1) \nabla u, \nabla u
\rangle_{\mathbb{R}^N} \,{\rm d}x }%
{ (p-1) \int_{\mathbb{R}^N} |u|^2\, \varphi_1^{p-2}\, m \,{\rm d}x }
\colon 0\not\equiv u\in \mathcal{D}_{\varphi_1}
\Big\} ,
\label{eq.lam_1}
\end{equation}
%
cf.\ eq.~\eqref{def.lam_1}.
This is a generalized Rayleigh quotient formula for
the first (smallest) eigenvalue of the selfadjoint operator
%
\begin{math}
(p-1)^{-1} \mathcal{A}_{\varphi_1} + \lambda_1\varphi_1^{p-2} m
\colon
\mathcal{D}_{\varphi_1} \to \mathcal{D}_{\varphi_1}' ,
\end{math}
%
where
$\mathcal{A}_{\varphi_1}$ has been defined in~\eqref{def.A}.
The following result determines all minimizers
for~\eqref{eq.lam_1}:
%
%%%%% UNIQUENESS and POSITIVITY - EV (Proposition)
%
\begin{proposition}\label{prop-Uni-EV}
Let\/ $1 0} be satisfied.
Then a function $u\in \mathcal{D}_{\varphi_1}$ satisfies
$\mathcal{Q}_0 (u,u) = 0$ if and only if
$u = \kappa\varphi_1$ for some constant $\kappa\in \mathbb{R}$.
%
\end{proposition}
%
The analogue of this proposition for a bounded domain
$\Omega\subset \mathbb{R}^N$ with a sufficiently regular boundary
$\partial\Omega$ is due to
{\sc Tak\'a\v{c}} \cite[Prop.\ 4.4, p.~202]{Takac-1}.
Our proof of Proposition~\ref{prop-Uni-EV} below
is a simplification of that given in \cite{Takac-1}. \smallskip
\begin{proof}{Proof of Proposition~\ref{prop-Uni-EV}}
Recall that the embedding
$\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$
is compact, by
Proposition \ref{prop-Compact}(b).
Let $u$ be any (nontrivial) minimizer for $\lambda_1$
in~\eqref{eq.lam_1}.
If $u$ changes sign in $\mathbb{R}^N$, denote
$u^+ = \max\{ u, 0\}$ and $u^- = \max\{ -u, 0\}$.
Then we have, using
{\sc Gilbarg} and {\sc Trudinger}
\cite[Theorem 7.8, p.~153]{GilbargTrud},
%
\begin{align*}
\lambda_1
& =
\frac{ \int_{\mathbb{R}^N} (u^+)^2\, \varphi_1^{p-2}\, m \,{\rm d}x }%
{ \int_{\mathbb{R}^N} u^2\, \varphi_1^{p-2}\, m \,{\rm d}x }
\cdot
\frac{ \int_{\mathbb{R}^N}
\langle {\bf A}(\nabla\varphi_1) \nabla u^+, \nabla u^+
\rangle_{\mathbb{R}^N} \,{\rm d}x }%
{ (p-1) \int_{\mathbb{R}^N} (u^+)^2\, \varphi_1^{p-2}\, m \,{\rm d}x }
\\
&\quad +
\frac{ \int_{\mathbb{R}^N} (u^-)^2\, \varphi_1^{p-2}\, m \,{\rm d}x }%
{ \int_{\mathbb{R}^N} u^2\, \varphi_1^{p-2}\, m \,{\rm d}x }
\cdot
\frac{ \int_{\mathbb{R}^N}
\langle {\bf A}(\nabla\varphi_1) \nabla u^-, \nabla u^-
\rangle_{\mathbb{R}^N} \,{\rm d}x }%
{ (p-1) \int_{\mathbb{R}^N} (u^-)^2\, \varphi_1^{p-2}\, m \,{\rm d}x }
\\
& \geq
\Big(
\frac{ \int_{\mathbb{R}^N} (u^+)^2\, \varphi_1^{p-2}\, m \,{\rm d}x }%
{ \int_{\mathbb{R}^N} u^2\, \varphi_1^{p-2}\, m \,{\rm d}x }
+ \frac{ \int_{\mathbb{R}^N} (u^-)^2\, \varphi_1^{p-2}\, m \,{\rm d}x }%
{ \int_{\mathbb{R}^N} u^2\, \varphi_1^{p-2}\, m \,{\rm d}x }
\Big) \lambda_1
= \lambda_1 .
\end{align*}
%
Consequently, both
$u^+$ and $u^-$ are (nontrivial) minimizers for $\lambda_1$.
Next, we show that if $u\in \ker(\mathcal{A}_{\varphi_1})$
then $u$ is a constant multiple of $\varphi_1$.
Since
$\varphi_1$ satisfies~\eqref{e:varphi.l_1},
it is of class $C^\infty$ in $\mathbb{R}^N\setminus \{ 0\}$,
by classical regularity theory
\cite[Theorem 8.10, p.~186]{GilbargTrud}.
Now, for each $\gamma\in \mathbb{R}$ fixed, consider the function
$v_\gamma\eqdef u - \gamma\varphi_1$ in $\mathbb{R}^N$.
Then both $v_\gamma^+$ and $v_\gamma^-$ belong to
$\ker(\mathcal{A}_{\varphi_1})$ and thus satisfy the equation
%
\begin{equation}
- \nabla\cdot
\left( {\bf A}(\nabla\varphi_1) \nabla v_\gamma^\pm \right)
= \lambda_1 (p-1) \varphi_1^{p-2} m v_\gamma^\pm \geq 0
\quad\mbox{in } \mathbb{R}^N\setminus \{ 0\} .
\label{A_phi=0:U}
\end{equation}
%
Again, we have
$v_\gamma^\pm \in C^\infty( \mathbb{R}^N\setminus \{ 0\} )$.
So we may apply the strong maximum principle
\cite[Theorem 3.5, p.~35]{GilbargTrud}
to eq.~\eqref{A_phi=0:U}
to conclude that either
$v_\gamma^+ \equiv 0$ in $\mathbb{R}^N\setminus \{ 0\}$, or else
$v_\gamma^+ > 0$ throughout $\mathbb{R}^N\setminus \{ 0\}$,
and similarly for $v_\gamma^-$.
This means that
$\mathop{\rm sign} (u - \gamma\varphi_1) \equiv \mathrm{const}$
in $\mathbb{R}^N\setminus \{ 0\}$.
Moving $\gamma$ from $-\infty$ to $+\infty$, we get
$u\equiv \kappa\varphi_1$ in $\mathbb{R}^N\setminus \{ 0\}$
for some constant $\kappa\in \mathbb{R}$.
This means
$u = \kappa\varphi_1$ in $\mathcal{D}_{\varphi_1}$, as claimed.
\end{proof}
\section{An improved Poincar\'e inequality ($2\leq p < N$)}
\label{s:Impr_Poinc}
We need a few more technical tools from
{\sc Fleckinger} and {\sc Tak\'a\v{c}} \cite[Sect.~5]{FleckTakac-1}
to prove Lemma~\ref{lem-Poincare}.
Although our present situation requires only a few changes
in the space setting in \cite{FleckTakac-1},
we provide complete proofs of all results
for the convenience of the reader.
%
%%%%% An Improved Poincare INEQUALITY (Remark)
%
\begin{remark}\label{rem-Poincare}
\begingroup\rm
Except when $u^\parallel = 0$, we may replace
$u\in D^{1,p}(\mathbb{R}^N)$ by $v = u / u^\parallel$
in inequality~\eqref{e:Poincare}
and thus restate it equivalently as follows, for all
$v^\top\in D^{1,p}(\mathbb{R}^N)$ with
$\int_{\mathbb{R}^N} v^\top\, \varphi_1^{p-1}\, m \,{\rm d}x = 0$:
%
\begin{equation}
\mathcal{Q}_{v^\top}(v^\top, v^\top)
= \mathcal{F}(\varphi_1 + v^\top)
\geq \frac{c}{p}
\left(
\| v^\top\|_{ \mathcal{D}_{\varphi_1} }^2
+ \| v^\top\|_{ D^{1,p}(\mathbb{R}^N) }^p
\right) .
\label{e:Poinc}
\end{equation}
%
\endgroup
\end{remark}
%
This remark indicates that
our proof of inequality~\eqref{e:Poincare}
should distinguish between the cases when the ratio
$\| u^\top\|_{ D^{1,p}(\mathbb{R}^N) } / | u^\parallel |$
is bounded away from zero by a constant $\gamma > 0$, say,
\[
\| u^\top\|_{ D^{1,p}(\mathbb{R}^N) } / | u^\parallel |
\geq \gamma ,
\]
and when it is sufficiently small, say,
\[
\| u^\top\|_{ D^{1,p}(\mathbb{R}^N) } / | u^\parallel |
\leq \gamma
\]
where $\gamma > 0$ is small enough.
The former case is treated in a standard way analogous
to~\eqref{def.lam_1},
whereas
the latter case requires a more sophisticated approach based on
the second\--order Taylor formula \eqref{def.Q}
applied to the expression
$\mathcal{Q}_{v^\top}(v^\top, v^\top)$
on the left\--hand side in~\eqref{e:Poinc}
where $v = u / u^\parallel$.
For either of these cases we need a separate auxiliary result:
We derive two formulas for Rayleigh quotients
outside and inside an arbitrarily small cone around
the axis spanned by $\varphi_1$, respectively.
%
\subsection{Minimization outside a cone around $\varphi_1$}
\label{ss:Ext-Cone}
We allow $1 < p < N$ throughout this paragraph.
Given any number $0 < \gamma < \infty$, we set
%
\begin{gather*}
\mathcal{C}_\gamma \eqdef
\Big\{
u\in D^{1,p}(\mathbb{R}^N) \colon
\| u^\top \|_{ D^{1,p}(\mathbb{R}^N) } \leq \gamma | u^\parallel |
\Big\} ,\\
\mathcal{C}_\gamma^\prime \eqdef
\Big\{
u\in D^{1,p}(\mathbb{R}^N) \colon
\| u^\top \|_{ D^{1,p}(\mathbb{R}^N) } \geq \gamma | u^\parallel |
\Big\} .
\end{gather*}
%
Note that $\mathcal{C}_\gamma$ is a closed cone in $D^{1,p}(\mathbb{R}^N)$ and
$\mathcal{C}_\gamma^\prime$ is the closure of $\mathcal{C}_\gamma^c$,
the complement of $\mathcal{C}_\gamma$ in $D^{1,p}(\mathbb{R}^N)$.
We consider also the hyperplane
\[
\mathcal{C}_\infty^\prime \eqdef
\Big\{
u\in D^{1,p}(\mathbb{R}^N) \colon u^\parallel = 0
\Big\}
= \bigcap_{ 0 < \gamma < \infty } \mathcal{C}_\gamma^\prime .
\]
For $0 < \gamma\leq \infty$ we define
%
\begin{equation}
\Lambda_\gamma \eqdef
\inf \Big\{
\frac{ \int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x }%
{ \int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x }
\colon u\in \mathcal{C}_\gamma^\prime \setminus \{ 0\}
\Big\} .
\label{def.Lam_g}
\end{equation}
%
The next result is an analogue of
\cite[Lemma 5.1, p.~963]{FleckTakac-1}
proved for a bounded domain $\Omega\subset \mathbb{R}^N$.
%%%%% Exterior Cone Minimum (Lemma)
\begin{lemma}\label{lem-Ext-Cone}
Let\/ $1 < p < N$ and\/ $0 < \gamma\leq \infty$.
Then we have $\Lambda_\gamma > \lambda_1$.
\end{lemma}
%
\begin{proof}
Assume the contrary, that is,
$\Lambda_\gamma = \lambda_1$ for some $0 < \gamma < \infty$.
Pick a minimizing sequence
$\{ u_n\}_{n=1}^\infty$ in $\mathcal{C}_\gamma^\prime$ such that
\[
\int_{\mathbb{R}^N} |u_n|^p\, m \,{\rm d}x = 1
\quad\mbox{and }\quad
\int_{\mathbb{R}^N} |\nabla u_n|^p \,{\rm d}x \to \lambda_1
\quad\mbox{as } n\to \infty .
\]
Since $D^{1,p}(\mathbb{R}^N)$ is a reflexive Banach space,
the minimizing sequence contains a weakly convergent subsequence in
$D^{1,p}(\mathbb{R}^N)$ which we denote by
$\{ u_n\}_{n=1}^\infty$ again.
Consequently,
$u_n\to u$ strongly in $L^p(\mathbb{R}^N; m)$, by
Proposition \ref{prop-Compact}(a), and
$\nabla u_n\rightharpoonup \nabla u$ weakly in $[ L^p(\mathbb{R}^N) ]^N$
as $n\to \infty$.
We deduce that
$\int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x = 1$ and
\[
\lambda_1^{1/p} \leq
\| \nabla u\|_{L^p(\mathbb{R}^N)} \leq
\liminf_{n\to \infty} \| \nabla u_n\|_{L^p(\mathbb{R}^N)}
= \lambda_1^{1/p} .
\]
As the standard norm on the space $D^{1,p}(\mathbb{R}^N)$
is uniformly convex,
by Clarkson's inequalities, we must have
$u_n\to u$ strongly in $D^{1,p}(\mathbb{R}^N)$,
by the proof of Milman's theorem
(see {\sc Yosida} \cite[Theorem V.2.2, p.~127]{Yosida}).
This means that
%
\begin{gather*}
u_n^\parallel
= \int_{\mathbb{R}^N} u_n\, \varphi_1^{p-1}\, m \,{\rm d}x
\to
u^\parallel
= \int_{\mathbb{R}^N} u\, \varphi_1^{p-1}\, m \,{\rm d}x , \\
u_n^\top = u_n - u_n^\parallel \varphi_1 \to
u^\top = u - u^\parallel \varphi_1
\mbox{ strongly in } D^{1,p}(\mathbb{R}^N) ,
\end{gather*}
%
as $n\to \infty$.
The set $\mathcal{C}_\gamma^\prime$ being closed in $D^{1,p}(\mathbb{R}^N)$,
we thus have
$u\in \mathcal{C}_\gamma^\prime$.
On the other hand, from
$\| u\|_{ L^p(\mathbb{R}^N; m) } = 1$ and
$\| \nabla u\|_{L^p(\mathbb{R}^N)} = \lambda_1^{1/p}$,
combined with the simplicity of the first eigenvalue $\lambda_1$,
one deduces that $u = \pm\varphi_1$, a contradiction to
$u\in \mathcal{C}_\gamma^\prime$.
The lemma is proved.
\end{proof}
%
\subsection{Minimization inside a cone around $\varphi_1$}
\label{ss:Int-Cone}
For $\phi\in D^{1,p}(\mathbb{R}^N)$, $\phi\not\equiv 0$ in $\mathbb{R}^N$,
let us define
%
\begin{equation}
\tilde\Lambda \eqdef
\liminf_{
\begin{smallmatrix}
{ \|\phi\|_{ D^{1,p}(\mathbb{R}^N) } \to 0 }\\
{ \langle\phi, \varphi_1^{p-1} m\rangle = 0 }
\end{smallmatrix}
}
\frac{
\int_{\mathbb{R}^N}
\left\langle
\left[
\int_0^1 \mathbf{A}
\left(
\nabla ( \varphi_1 + s\phi )
\right) (1-s) \,{\rm d}s
\right] \nabla\phi ,\, \nabla\phi
\right\rangle_{\mathbb{R}^N} \,{\rm d}x
}{
\int_{\mathbb{R}^N}
\left[
\int_0^1 |\varphi_1 + s\phi|^{p-2} (1-s) \,{\rm d}s
\right] |\phi|^2\, m \,{\rm d}x
}
\label{def.Lam_0}
\end{equation}
%
with the abbreviation~\eqref{def.A=F''}.
Using the quadratic form $\mathcal{Q}_\phi$ defined
in~\eqref{def.Q}, we notice that
\[
\tilde\Lambda - \lambda_1 (p-1) =
\liminf_{
\begin{smallmatrix}
{ \|\phi\|_{ D^{1,p}(\mathbb{R}^N) } \to 0 }\\
{ \langle\phi, \varphi_1^{p-1} m\rangle = 0 }
\end{smallmatrix}
}
\frac{ \mathcal{Q}_{\phi}(\phi,\phi)
}{
\int_{\mathbb{R}^N}
\left[
\int_0^1 |\varphi_1 + s\phi|^{p-2} (1-s) \,{\rm d}s
\right] |\phi|^2\, m \,{\rm d}x
} \geq 0 .
\]
The next result parallels
\cite[Lemma 5.2, p.~964]{FleckTakac-1}
shown for a bounded domain $\Omega\subset \mathbb{R}^N$.
%%%%% Interior Cone Minimum (Lemma)
\begin{lemma}\label{lem-Int-Cone}
Let\/ $2\leq p < N$.
We have $\tilde\Lambda > \lambda_1 (p-1)$.
%
\end{lemma}
%
Before giving the proof of this inequality,
we first recall that the kernels of the quadratic forms
$\mathcal{Q}_{\phi}(v,v)$ and
$\mathcal{Q}_0(v,v)$ defined in
\eqref{def.Q} and \eqref{def.Q_0}, respectively,
can be compared by inequalities
\eqref{1-s.A.geom:p>2} for $p\geq 2$, and
\eqref{1-s.A.geom:p<2} for $p<2$,
so that we can use the Hilbert space $\mathcal{D}_{\varphi_1}$
not only for $\mathcal{Q}_0$ but also for $\mathcal{Q}_{\phi}$.
Next, we introduce the following notations where
$t\in \mathbb{R}$ and $\phi\in D^{1,p}(\mathbb{R}^N)$:
%
\begin{gather*}
\mathcal{P}_0(t,\phi) \eqdef
\int_{\mathbb{R}^N}
\Big[ \int_0^1
| \varphi_1 + s t\phi |^{p-2} (1-s) \,{\rm d}s
\Big] \phi^2\, m \,{\rm d}x ,\\
\mathcal{P}_1(t,\phi) \eqdef
\int_{\mathbb{R}^N}
\Big\langle
\Big[
\int_0^1 \mathbf{A}
(
\nabla ( \varphi_1 + s t\phi )
) (1-s) \,{\rm d}s
\Big] \nabla\phi ,\, \nabla\phi
\Big\rangle_{\mathbb{R}^N} \,{\rm d}x .
\end{gather*}
%
Hence, equation~\eqref{def.Lam_0} takes the form
\[
\tilde\Lambda =
\liminf_{
\begin{smallmatrix}
{ \|\phi\|_{ D^{1,p}(\mathbb{R}^N) } \to 0 }\\
{ \langle\phi, \varphi_1^{p-1} m\rangle = 0 }
\end{smallmatrix}
}
\frac{ \mathcal{P}_1(t,\phi) }{ \mathcal{P}_0(t,\phi) }
\quad\mbox{with any fixed } t\in \mathbb{R}\setminus \{ 0\} .
\]
Furthermore, due to inequalities \eqref{1-s.A.geom:p>2},
the expressions
$\mathcal{P}_0(t,\phi)$ and $\mathcal{P}_1(t,\phi)$,
respectively, are equivalent to
%
\begin{align*}
\mathcal{N}_0(t,\phi) &\eqdef
\int_{\mathbb{R}^N}
\left(
\varphi_1^{p-2} + |t|^{p-2} |\phi|^{p-2}
\right) \phi^2\, m \,{\rm d}x
\\
& = \int_{\mathbb{R}^N} \varphi_1^{p-2} \phi^2\, m \,{\rm d}x
+ |t|^{p-2} \|\phi\|_{ L^p(\mathbb{R}^N; m) }^p
\end{align*}
%
and
%
\begin{align*}
\mathcal{N}_1(t,\phi) &\eqdef
\int_{\mathbb{R}^N}
\left(
|\nabla\varphi_1|^{p-2}
+ |t|^{p-2} |\nabla\phi|^{p-2}
\right) |\nabla\phi|^2 \,{\rm d}x
\\
& = \|\phi\|_{ \mathcal{D}_{\varphi_1} }^2
+ |t|^{p-2} \|\phi\|_{ D^{1,p}(\mathbb{R}^N) }^p ,
\end{align*}
%
that is, there are two constants
$c_1, c_2 > 0$ independent from $t$ and $\phi$ such that
%
\begin{equation}
c_1\, \mathcal{N}_i(t,\phi) \leq \mathcal{P}_i(t,\phi) \leq
c_2\, \mathcal{N}_i(t,\phi) ;\quad i=0,1 .
\label{ineq.PN}
\end{equation}
%
\begin{proof}[Proof of Lemma~\ref{lem-Int-Cone}]
On the contrary, assume that
$\tilde\Lambda \leq \lambda_1 (p-1)$.
Pick a minimizing sequence
$\{ \phi_n\}_{n=1}^\infty$ in $D^{1,p}(\mathbb{R}^N)$
such that
$\phi_n\not\equiv 0$ in $\mathbb{R}^N$,
$\langle\phi_n, \varphi_1^{p-1} m\rangle = 0$,
$\|\phi_n\|_{ D^{1,p}(\mathbb{R}^N) } \to 0$,
and
\[
\frac{ \mathcal{P}_1(1,\phi_n) }{ \mathcal{P}_0(1,\phi_n) }
\,\longrightarrow \tilde\Lambda \leq \lambda_1 (p-1)
\quad\mbox{as } n\to \infty .
\]
Next, set
$t_n = \mathcal{P}_0(1,\phi_n)^{1/2}$
and
$V_n = \phi_n / t_n$ for $n=1,2,\dots$.
Hence, we have \hbox{$t_n\to 0$},
$\mathcal{P}_0(t_n,V_n) = 1$, and
$\mathcal{P}_1(t_n,V_n) \to \tilde\Lambda$ as $n\to \infty$.
Inequalities~\eqref{ineq.PN} guarantee that both sequences
$\| V_n\|_{ \mathcal{D}_{\varphi_1} }$ and
$t_n^{1 - (2/p)} \| V_n\|_{ D^{1,p}(\mathbb{R}^N) }$
are bounded, and so we may extract a subsequence
denoted again by $\{ V_n\}_{n=1}^\infty$ such that
$V_n\rightharpoonup V$ weakly in $\mathcal{D}_{\varphi_1}$ and
$t_n^{1 - (2/p)} V_n$ $\rightharpoonup z$ weakly in
$D^{1,p}(\mathbb{R}^N)$ as $n\to \infty$.
Using the imbedding
$D^{1,p}(\mathbb{R}^N) \hookrightarrow \mathcal{D}_{\varphi_1}$,
we get $z\equiv 0$ in $\mathbb{R}^N$.
Furthermore, both imbeddings
$D^{1,p}(\mathbb{R}^N) \hookrightarrow L^p(\mathbb{R}^N; m)$ and
$\mathcal{D}_{\varphi_1} \hookrightarrow \mathcal{H}_{\varphi_1}$
being compact by Proposition \ref{prop-Compact},
we have also
$V_n\to V$ strongly in $\mathcal{H}_{\varphi_1}$ and
$t_n^{1 - (2/p)} V_n\to 0$ strongly in $L^p(\mathbb{R}^N; m)$.
It follows that
$\langle V, \varphi_1^{p-1} m\rangle = 0$ and
%
\begin{gather*}
\mathcal{P}_0(0,V) = \frac12
\int_{\mathbb{R}^N} \varphi_1^{p-2} V^2 \,{\rm d}x = 1 , \\
\mathcal{P}_1(0,V) = \frac12
\left\langle
\mathbf{A} (\nabla\varphi_1) \nabla V ,\, \nabla V
\right\rangle \leq \tilde\Lambda \leq \lambda_1 (p-1) .
\end{gather*}
%
Consequently, Proposition~\ref{prop-Uni-EV} forces
$V = \kappa\varphi_1$ in $\mathbb{R}^N$,
where $\kappa\in \mathbb{R}$ is a constant,
$\kappa\neq 0$ by $\mathcal{P}_0(0,V) = 1$.
But this is a contradiction to
$\langle V, \varphi_1^{p-1} m\rangle = 0$.
We conclude that
$\tilde\Lambda > \lambda_1 (p-1)$ as claimed.
\end{proof}
\subsection{Proof of Lemma~\ref{lem-Poincare}}
\label{ss:Poincare}
If $u\in D^{1,p}(\mathbb{R}^N)$ satisfies
$\langle u, \varphi_1\rangle = 0$,
then equation~\eqref{def.Lam_g} implies
%
\begin{equation}
\begin{split}
\int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x
- \lambda_1 \int_{\mathbb{R}^N} |u|^p\, m \,{\rm d}x
& \geq
\big( 1 - \frac{\lambda_1}{\Lambda_\infty} \big)
\int_{\mathbb{R}^N} |\nabla u|^p \,{\rm d}x\\
&= \big( 1 - \frac{\lambda_1}{\Lambda_\infty} \big)
\int_{\mathbb{R}^N} |\nabla u^\top|^p \,{\rm d}x
\end{split}
\label{ineq:Lam_inf}
\end{equation}
%
where $\lambda_1 / \Lambda_\infty < 1$
by Lemma~\ref{lem-Ext-Cone}.
Thus, we may assume
$\langle u, \varphi_1\rangle \not= 0$
and so we need to prove only inequality~\eqref{e:Poinc}.
We will apply Lemmas \ref{lem-Ext-Cone} and~\ref{lem-Int-Cone}
to the following two cases, respectively.
\noindent {\it Case\/}
$\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) } \geq \gamma$:
Here, $\gamma > 0$ is an arbitrary, but fixed number.
In analogy with inequality~\eqref{ineq:Lam_inf} above,
we have
%
\begin{equation}
\begin{split}
& \int_{\mathbb{R}^N} |\nabla\varphi_1 + \nabla v^\top|^p \,{\rm d}x
- \lambda_1
\int_{\mathbb{R}^N} |\varphi_1 + v^\top|^p\, m \,{\rm d}x
\\
& \geq
\big( 1 - \frac{\lambda_1}{\Lambda_\gamma} \big)
\int_{\mathbb{R}^N} |\nabla\varphi_1 + \nabla v^\top|^p \,{\rm d}x
\geq c_\gamma
\int_{\mathbb{R}^N} |\nabla v^\top|^p \,{\rm d}x
\end{split}
\label{ineq:Lam_g}
\end{equation}
%
for all $v^\top\in D^{1,p}(\mathbb{R}^N)$ such that
$\langle v^\top, \varphi_1^{p-1} m\rangle = 0$
and
$\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) } \geq \gamma$,
where $c_\gamma > 0$ is a constant independent from $v^\top$.
The last inequality follows from the boundedness of
the orthogonal projections
$u\mapsto u^\parallel\cdot \varphi_1$ and
$u\mapsto u^\top$ in $D^{1,p}(\mathbb{R}^N)$.
Recalling the imbedding
$D^{1,p}(\mathbb{R}^N) \hookrightarrow \mathcal{D}_{\varphi_1}$,
we deduce from~\eqref{ineq:Lam_g}
that inequality~\eqref{e:Poinc} is valid provided
$\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) }$ $\geq \gamma$.
\noindent {\it Case\/}
$\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) } \leq \gamma$:
Here, $\gamma > 0$ is sufficiently small.
According to equation~\eqref{def.Lam_0} and
Lemma~\ref{lem-Int-Cone} we have
%
\begin{equation}
\begin{split}
\mathcal{Q}_{v^\top}(v^\top, v^\top)
&= \mathcal{P}_1(1,v^\top)
- \lambda_1 (p-1)\, \mathcal{P}_0(1,v^\top)
\\
& \geq
\Big( 1 - \frac{\lambda_1 (p-1)}{\tilde\Lambda}
\Big)
\mathcal{P}_1(1,v^\top)\\
&\geq \tilde c\cdot \mathcal{N}_1(1,v^\top)
\end{split}
\label{ineq:Lam_0}
\end{equation}
%
for all $v^\top\in D^{1,p}(\mathbb{R}^N)$ such that
$\langle v^\top, \varphi_1^{p-1} m\rangle = 0$
and
$\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) } \leq \gamma$,
where $\gamma > 0$ is sufficiently small and
$\tilde c > 0$ is a constant independent from $v^\top$.
Recall that the expressions
$\mathcal{P}_i(1,v^\top)$ and
$\mathcal{N}_i(1,v^\top)$ ($i=0,1$)
have been defined after Lemma~\ref{lem-Int-Cone}.
From~\eqref{ineq:Lam_0} we deduce that
inequality~\eqref{e:Poinc} is valid also when
$\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) } \leq \gamma$.
%\end{proof}
%%%%% Coercivity of J_{\lambda_1} (Remark)
\begin{remark}\label{rem-E_f:coercive}
\begingroup\rm
Assume $2 < p < N$ and let
$f\in \mathcal{D}_{\varphi_1}'$ satisfy
$\langle f,\varphi_1\rangle = 0$.
Recall that
$D^{1,p}(\mathbb{R}^N) \hookrightarrow \mathcal{D}_{\varphi_1}$.
Although the functional $\mathcal{J}_{\lambda_1}$,
defined in \eqref{def.jl} with $\lambda = \lambda_1$,
is no longer coercive on $D^{1,p}(\mathbb{R}^N)$,
it is still not only bounded from below, but also
``very close'' to being coercive on the weighted Sobolev space
$\mathcal{D}_{\varphi_1}$,
as a direct consequence of improved Poincar\'e's inequality
\eqref{e:Poincare}.
This property of $\mathcal{J}_{\lambda_1}$
will be used in the next section to prove the existence theorem
(Theorem~\ref{thm-Exist:p>2})
for problem \eqref{e:BVP.l_1}.
\endgroup
\end{remark}
%
\section{Proof of Theorem~\ref{thm-Exist:p>2}}
\label{s:pr-Exist:p>2}
Our proof of Theorem~\ref{thm-Exist:p>2}
combines the improved Poincar\'e inequality~\eqref{e:Poincare}
with a generalized Rayleigh quotient formula.
To this end, we may assume that
$f\in \mathcal{D}_{\varphi_1}'$ satisfies
$f\not\equiv 0$ in $\mathbb{R}^N$ and
$\langle f, \varphi_1\rangle = 0$.
Define the number $M_f$, for $0\leq M_f\leq \infty$, by
%
\begin{equation}
M_f\eqdef
\sup_{
\begin{smallmatrix}
v\in D^{1,p}(\mathbb{R}^N)\\
v\not\in \{ \kappa\varphi_1\colon \kappa\in \mathbb{R}\}
\end{smallmatrix}
}
\frac{ | \langle f, v\rangle |^p }%
{
\int_{\mathbb{R}^N} |\nabla v|^p \,{\rm d}x
- \lambda_1
\int_{\mathbb{R}^N} |v|^p\, m \,{\rm d}x
}\, .
\label{def.M_f}
\end{equation}
%
Clearly, $M_f > 0$.
Moreover,
inequality~\eqref{e:Poincare} entails
%
\[
| \langle f, v\rangle |^p
\leq \| f\|_{ D^{-1,p'}(\mathbb{R}^N) }^p\,
\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) }^p
\leq C_f
\Big(
\int_{\mathbb{R}^N} |\nabla v|^p \,{\rm d}x
- \lambda_1
\int_{\mathbb{R}^N} |v|^p\, m \,{\rm d}x
\Big)
\]
%
for all $v\in D^{1,p}(\mathbb{R}^N)$, where
$C_f = c^{-1}\, \| f\|_{ D^{-1,p'}(\mathbb{R}^N) }^p$
is a constant.
This shows that $M_f\leq C_f < \infty$.
In a similar way we arrive at
%
\begin{equation}
\begin{split}
| v^\parallel |^{p-2}\,
| \langle f, v\rangle |^2
&\leq | v^\parallel |^{p-2}
\left( \| f\|_{ \mathcal{D}_{\varphi_1}' } \right)^2\,
\| v^\top\|_{ \mathcal{D}_{\varphi_1} }^2 \\
& \leq C_f'
\Big(
\int_{\mathbb{R}^N} |\nabla v|^p \,{\rm d}x
- \lambda_1
\int_{\mathbb{R}^N} |v|^p\, m \,{\rm d}x
\Big)
\quad\mbox{for all } v\in D^{1,p}(\mathbb{R}^N) ,
\end{split}
\label{ineq:f.v_2}
\end{equation}
%
where
$C_f'= c^{-1} ( \| f\|_{ \mathcal{D}_{\varphi_1}' } )^2$
is a constant, and
$\|\cdot\|_{ \mathcal{D}_{\varphi_1}' }$
stands for the dual norm on $\mathcal{D}_{\varphi_1}'$.
>From \eqref{def.M_f} and inequality~\eqref{ineq:f.v_2}
we can draw the following conclusion:
If $v\in D^{1,p}(\mathbb{R}^N)$ is such that
$v^\top \not\equiv 0$ in $\mathbb{R}^N$ and
\[
\frac{ | \langle f, v\rangle |^p }%
{
\int_{\mathbb{R}^N} |\nabla v|^p \,{\rm d}x
- \lambda_1
\int_{\mathbb{R}^N} |v|^p\, m \,{\rm d}x
}
\geq \frac{1}{2} M_f ,
\]
then
$\langle f, v\rangle \neq 0$ and
\[
| v^\parallel |^{p-2}
\leq 2 (C_f' / M_f)\,
| \langle f, v\rangle |^{p-2}
\leq (C_f'')^{p-2}\,
\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) }^{p-2} ,
\]
where
%
\begin{math}
C_f''=
[ 2 (C_f' / M_f) ]^{1/(p-2)}\, \| f\|_{ D^{-1,p'}(\mathbb{R}^N) }
\end{math}
%
is a constant, i.e.,
%
\begin{equation}
| v^\parallel |
\leq C_f''\,
\| v^\top\|_{ D^{1,p}(\mathbb{R}^N) } .
\label{ineq:v_f.v}
\end{equation}
%
Next, take any maximizing sequence
$\{ v_n\}_{n=1}^\infty$ in $D^{1,p}(\mathbb{R}^N)$
for the generalized Rayleigh quotient~\eqref{def.M_f},
that is, $v_n^\top \not\equiv 0$ in $\mathbb{R}^N$ and
%
\begin{equation}
\frac{ | \langle f, v_n\rangle |^p }%
{
\int_{\mathbb{R}^N} |\nabla v_n|^p \,{\rm d}x
- \lambda_1
\int_{\mathbb{R}^N} |v_n|^p\, m \,{\rm d}x
}\,\longrightarrow M_f
\quad\mbox{as } n\to \infty .
\label{M_f:v_n}
\end{equation}
%
Since both, the numerator and the denominator are $p$-homogeneous,
we may assume
$\| v_n\|_{ D^{1,p}(\mathbb{R}^N) } = 1$ for all $n\geq 1$.
The Sobolev space $D^{1,p}(\mathbb{R}^N)$ being reflexive,
we may pass to a convergent subsequence
$v_n\rightharpoonup w$ weakly in $D^{1,p}(\mathbb{R}^N)$;
hence, also $v_n\to w$ strongly in $L^p(\mathbb{R}^N; m)$, by
Proposition \ref{prop-Compact}(a), and
$\langle f, v_n\rangle \to \langle f, w\rangle$
as $n\to \infty$.
We insert these limits into~\eqref{M_f:v_n} to obtain
%
\begin{equation}
\int_{\mathbb{R}^N} |\nabla w|^p \,{\rm d}x
- \lambda_1
\int_{\mathbb{R}^N} |w|^p\, m \,{\rm d}x
\leq 1 - \lambda_1
\int_{\mathbb{R}^N} |w|^p\, m \,{\rm d}x
= M_f^{-1}
| \langle f, w\rangle |^p .
\label{M_f:w}
\end{equation}
%
In particular, we have $w\not\equiv 0$ in $\mathbb{R}^N$,
therefore also $w^\top \not\equiv 0$
by~\eqref{ineq:v_f.v}, and consequently
$| \langle f, w\rangle | \not= 0$
by~\eqref{M_f:w}.
We combine \eqref{def.M_f} with~\eqref{M_f:w} to get
$\int_{\mathbb{R}^N} |\nabla w|^p \,{\rm d}x = 1$.
Hence, the supremum $M_f$ in~\eqref{def.M_f}
is attained at $w$ in place of $v$.
Finally, we can apply
the calculus of variations to the inequality
%
\[
\int_{\mathbb{R}^N} |\nabla v|^p \,{\rm d}x
- \lambda_1
\int_{\mathbb{R}^N} |v|^p\, m \,{\rm d}x
- M_f^{-1}
| \langle f, v\rangle |^p
\geq 0
\quad\mbox{for } v\in D^{1,p}(\mathbb{R}^N)
\]
%
to derive
%
\[
- \Delta_p w - \lambda_1\, m\, |w|^{p-2} w
= M_f^{-1}
| \langle f, w\rangle |^{p-2}
\langle f, w\rangle \cdot f(x)
\quad\mbox{in } \mathbb{R}^N .
\]
%
It follows that
$u\eqdef M_f^{1/(p-1)} \langle f, w\rangle^{-1} \cdot w$
is a weak solution of problem~\eqref{e:BVP.l_1}.
Theorem~\ref{thm-Exist:p>2} is proved.
%\end{proof}
\section{Proof of Theorem~\ref{thm-Exist:p<2}}
\label{s:pr-Exist:p<2}
In contrast to the case $2\leq p < N$ in Section~\ref{s:Impr_Poinc},
Remark~\ref{rem-E_f:coercive},
for $1 \lambda_1$ in formula \eqref{def.Lam_g}.
This shows that the functional $\mathcal{E}_f$ is coercive on
$\mathcal{C}_\infty^\prime = D^{1,p}(\mathbb{R}^N)^\top$.
Hence, being also weakly lower semicontinuous,
$\mathcal{E}_f$ possesses a global minimizer $u_0^\top$ over
$D^{1,p}(\mathbb{R}^N)^\top$,
\[
\mathcal{E}_f(u_0^\top)
= \inf_{ w\in D^{1,p}(\mathbb{R}^N)^\top } \mathcal{E}_f(w)
> -\infty .
\]
Now let us look for the functions $u$ and $v$, respectively,
in Definition \ref{def-Saddle_Geom} in the forms of
%
\begin{equation}
u_\pm = \pm\tau \varphi_1 + \tau^{1 - (p/2)} \phi
\quad\mbox{with $\tau\in (0,\infty)$ sufficiently large, }
\label{def.u,v}
\end{equation}
%
where $\phi\in C_{\mathrm{c}}^1(\mathbb{R}^N)$
is a function chosen as follows:
%
\begin{itemize}
%
\item[($\boldsymbol{\Phi}$)]
$\langle f, \phi\rangle = 1$ and $0\not\in K$ where
%
\begin{equation*}
K = \mathop{\rm supp}(\phi)\eqdef
\overline{ \{ x\in \mathbb{R}^N\colon \phi(x)\not= 0\} }
\quad ( \subset \mathbb{R}^N )
\end{equation*}
%
denotes the support of $\phi$.
%
\end{itemize}
%
The existence of $\phi$ is verified as follows.
Since
$f\in D^{-1,p'}(\mathbb{R}^N)$ satisfies
$f\not\equiv 0$ in $\mathbb{R}^N$, there is a function
$\phi_0\in C_{\mathrm{c}}^1(\mathbb{R}^N)$ such that
$\langle f, \phi_0\rangle = 1$.
On the contrary to ($\boldsymbol{\Phi}$),
suppose that the support
$K_0 = \mathop{\rm supp}(\phi_0)$ of $\phi_0$ always contains $0\in \mathbb{R}^N$.
This is equivalent to saying that
$\langle f, \phi\rangle = 0$ whenever
$\phi\in C_{\mathrm{c}}^1(\mathbb{R}^N)$ is such that
$0\not\in \mathop{\rm supp}(\phi)$.
Now choose a $C^1$ function
$\psi\colon \mathbb{R}_+\to [0,1]$ such that
$\psi(r) = 1$ if $0\leq r\leq 1$,
$0\leq \psi(r)\leq 1$ if $1\leq r\leq 2$, and
$\psi(r) = 0$ if $2\leq r < \infty$.
Define
$\psi_n(x)\eqdef \psi(n|x|)$ for all $x\in \mathbb{R}^N$; $n=1,2,\dots$.
Then
$0\not\in \mathop{\rm supp}( (1-\psi_n)\phi_0 )$ which yields
$\langle f, (1-\psi_n)\phi_0\rangle = 0$.
Hence
%
\begin{math}
\langle f, \psi_n\phi_0\rangle = \langle f, \phi_0\rangle = 1 .
\end{math}
%
However, this is contradicted by
$\| \psi_n\phi_0\|_{ D^{1,p}(\mathbb{R}^N) } \to 0$
as $n\to \infty$, which follows easily from
%
\begin{equation*}
\| \nabla(\psi_n\phi_0) \|_{ L^p(\mathbb{R}^N) }
\leq \| \phi_0\|_{ L^\infty(\mathbb{R}^N) }
\| \nabla\psi_n\|_{ L^p(\mathbb{R}^N) }
+ \| \nabla\phi_0\|_{ L^\infty(\mathbb{R}^N) }
\| \psi_n\|_{ L^p(\mathbb{R}^N) }
\end{equation*}
%
with both
%
\begin{gather*}
\| \nabla\psi_n\|_{ L^p(\mathbb{R}^N) }
= n^{1 - (N/p)} \| \nabla\psi\|_{ L^p(\mathbb{R}^N) } \to 0,\\
\| \psi_n\|_{ L^p(\mathbb{R}^N) }
= n^{- (N/p)} \| \psi\|_{ L^p(\mathbb{R}^N) } \to 0
\end{gather*}
%
as $n\to \infty$, by $1 < p < 2\leq N$.
So let $\phi\in C_{\mathrm{c}}^1(\mathbb{R}^N)$
satisfy condition ($\boldsymbol{\Phi}$).
%
For $\tau\in (0,\infty)$ we compute
%
\begin{equation}
\int_{\mathbb{R}^N} u_\pm\, \varphi_1^{p-1}\, m \,{\rm d}x
= \pm\tau
+ \tau^{1 - (p/2)}
\int_{\mathbb{R}^N} \phi\, \varphi_1^{p-1}\, m \,{\rm d}x ,
\label{u_pm.phi_1}
\end{equation}
%
by
$\int_{\mathbb{R}^N} \varphi_1^p\, m \,{\rm d}x = 1$.
It follows that
%
\[
\int_{\mathbb{R}^N} u_-\, \varphi_1^{p-1}\, m \,{\rm d}x < 0 <
\int_{\mathbb{R}^N} u_+\, \varphi_1^{p-1}\, m \,{\rm d}x
\quad\mbox{for all $\tau > 0$ large enough.}
\]
%
Next we use eqs.\ \eqref{J''.phi_1} and \eqref{def.Q} together with
$\langle f, \varphi_1\rangle = 0$ to obtain
%
\begin{equation}
\begin{split}
& \mathcal{E}_f(u_\pm)
= \mathcal{J}_{\lambda_1}
( \pm\tau \varphi_1 + \tau^{1 - (p/2)} \phi )
=
\\
& \mathcal{Q}_{ \pm \tau^{-p/2} \phi }(\phi,\phi)
- \tau^{1 - (p/2)}\, \langle f, \phi\rangle
= \mathcal{Q}_{ \pm \tau^{-p/2} \phi }(\phi,\phi)
- \tau^{1 - (p/2)} .
\end{split}
\label{E_f:u_pm}
\end{equation}
%
We recall that the quadratic forms
$\mathcal{Q}_{ \pm\tau^{-p/2} \phi }$ are given by
formula \eqref{def.Q}.
Since
$\inf_K |\nabla\varphi_1|$ $> 0$, $\inf_K \varphi_1 > 0$,
and $\phi$ is supported in $K\subset \mathbb{R}^N\setminus \{ 0\}$,
we conclude that both summands in
$\mathcal{Q}_{ \pm\tau^{-p/2} \phi }(\phi,\phi)$
are bounded independently from $\tau\geq \tau_0$,
provided $\tau_0\in (0,\infty)$ is large enough.
Finally, from \eqref{E_f:u_pm} we deduce that
$\mathcal{E}_f(u_\pm) \to -\infty$ as $\tau\to +\infty$.
The conclusion of the lemma follows.
\end{proof}
%
\subsection{A minimax method}
\label{ss:Minimax}
We allow $1 0 .
\]
%
Note that for any fixed $\tau\in \mathbb{R}$ the functional
$u^\top \mapsto \mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top )$
is coercive on the (closed linear) subspace
$D^{1,p}(\mathbb{R}^N)^\top$ of $D^{1,p}(\mathbb{R}^N)$.
This claim follows from the following inequalities
which are valid whenever
%
\begin{math}
|\tau|\leq T\leq
\gamma_\eta^{-1} \| u^\top\|_{ D^{1,p}(\mathbb{R}^N) } ,
\end{math}
%
for any fixed $T\in (0,\infty)$:
%
\begin{equation}
\begin{split}
& \int_{\mathbb{R}^N} |\nabla ( \tau\varphi_1 + u^\top )|^p \,{\rm d}x
- (\Lambda_\infty - \eta)
\int_{\mathbb{R}^N} |( \tau\varphi_1 + u^\top )|^p\, m \,{\rm d}x
\\
& \geq \Big(
1 - \frac{ \Lambda_\infty - \eta }{ \Lambda_{\gamma_\eta} }
\Big)
\int_{\mathbb{R}^N} |\nabla ( \tau\varphi_1 + u^\top )|^p \,{\rm d}x
\\
& \geq \Big(
1 - \frac{ \Lambda_\infty - \eta }{ \Lambda_{\gamma_\eta} }
\Big)
\left|
\| \nabla u^\top\|_{ L^p(\mathbb{R}^N) }
- |\tau|\cdot \| \nabla\varphi_1\|_{ L^p(\mathbb{R}^N) }
\right|^p
\\
& \geq c\, \| \nabla u^\top\|_{ L^p(\mathbb{R}^N) }^p
- c_T ,
\end{split}
\label{ineq:Lam_1.T}
\end{equation}
%
with another constant $0 < c_T < \infty$ depending solely on $T$.
The first inequality in~\eqref{ineq:Lam_1.T}
is easily derived from formula \eqref{def.Lam_g}.
Consequently, any global minimizer $u_\tau^\top$
for the functional
$u^\top \mapsto \mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top )$
on $D^{1,p}(\mathbb{R}^N)^\top$ satisfies the estimate
$\| u_\tau^\top\|_{ D^{1,p}(\mathbb{R}^N) } \leq C_T$ $< \infty$,
where $C_T$ is a constant independent from
$\lambda\in [ 0, \Lambda_\infty - \eta ]$ and $\tau\in [-T,T]$.
Such a global minimizer always exists and verifies
the Euler\--Lagrange equation
%
\begin{equation}
\begin{split}
&- \Delta_p( \tau\varphi_1 + u_\tau^\top )
- \lambda\, m(x)\,
| \tau\varphi_1 + u_\tau^\top |^{p-2}
( \tau\varphi_1 + u_\tau^\top ) \\
& = f^\top(x) + \zeta_\tau\cdot m(x)\, \varphi_1(x)^{p-1}
\quad\mbox{in } \mathbb{R}^N ,
\end{split}
\label{t:BVP.u_tau}
\end{equation}
%
with a Lagrange multiplier $\zeta_\tau\in \mathbb{R}$.
Thus, we may define
%
\begin{equation}
j_\lambda(\tau)\eqdef
\min_{ u^\top\in D^{1,p}(\mathbb{R}^N)^\top }
\mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top ) .
\label{def.j_lam,tau}
\end{equation}
%
In the rest of our proof of Theorem \ref{thm-Exist:p<2}
in {\S}\ref{ss:Fredh_Mount}
we will show that for $1 0 .
\label{j_disc}
\end{equation}
%
Consider any global minimizer $u_n^\top$
for the functional
%
\begin{math}
u^\top \mapsto
\mathcal{J}_{\mu_n}( \tau_n\varphi_1 + u^\top ; f_n )
\end{math}
%
on $D^{1,p}(\mathbb{R}^N)^\top$; $n=1,2,\dots$.
The sequence
$\{ u_n^\top \}_{n=1}^\infty$
is bounded in $D^{1,p}(\mathbb{R}^N)$, by ineq.~\eqref{ineq:Lam_1.T},
and hence, it contains a weakly convergent subsequence
(indexed by $n$ again)
$u_n^\top \rightharpoonup w^\top$ in $D^{1,p}(\mathbb{R}^N)^\top$
as $n\to \infty$.
>From the weak lower semicontinuity of
$\mathcal{J}_{\lambda}$ on $D^{1,p}(\mathbb{R}^N)$ we obtain
%
\begin{equation}
\begin{split}
\liminf_{n\to \infty} j_{\mu_n}(\tau_n;f_n)
&= \liminf_{n\to \infty}
\mathcal{J}_{\mu_n}( \tau_n\varphi_1 + u_n^\top ; f_n )
\\
& \geq \mathcal{J}_{\mu_0}( \tau_0\varphi_1 + w^\top ; f_0 )
\geq j_{\mu_0}(\tau_0;f_0) .
\end{split}
\label{j_lower}
\end{equation}
%
On the other hand, if $u_0^\top$ is any global minimizer
for the functional
$u^\top \mapsto$
$\mathcal{J}_{\mu_0}( \tau_0\varphi_1 + u^\top ; f_0 )$
on $D^{1,p}(\mathbb{R}^N)^\top$, then one has
%
\begin{equation}
\begin{split}
\limsup_{n\to \infty} j_{\mu_n}(\tau_n;f_n)
& \leq \lim_{n\to \infty}
\mathcal{J}_{\mu_n}( \tau_n\varphi_1 + u_0^\top ; f_n )
\\
& = \mathcal{J}_{\mu_0}( \tau_0\varphi_1 + u_0^\top ; f_0 )
= j_{\mu_0}(\tau_0;f_0) .
\end{split}
\label{j_upper}
\end{equation}
%
We combine inequalities \eqref{j_lower} and~\eqref{j_upper}
to get
\[
\lim_{n\to \infty} j_{\mu_n}(\tau_n;f_n) = j_{\mu_0}(\tau_0;f_0)
\]
which contradicts~\eqref{j_disc}.
The continuity of
$(\tau,\lambda,f) \mapsto j_\lambda(\tau;f)$ is proved.
Finally, the equicontinuity of the family~\eqref{j_equicont}
is a consequence of the uniform continuity of
the mapping~\eqref{j_cont} on the compact set
$[-T,T]\times [0, \Lambda_\infty - \eta]\times K$.
\end{proof}
%%%%% Continuity of Constrained Minimum (Remark)
\begin{remark}\label{rem-Cont-Min}
\begingroup\rm
We claim that in the proof of Lemma~\ref{lem-Cont-Min},
$w^\top$ is a global minimizer for the functional
%
\begin{math}
u^\top \mapsto
\mathcal{J}_{\mu_0}( \tau_0\varphi_1 + u^\top ; f_0 )
\end{math}
%
on $D^{1,p}(\mathbb{R}^N)^\top$ and we have also
$u_n^\top \to w^\top$ strongly in $D^{1,p}(\mathbb{R}^N)$
as $n\to \infty$.
First of all,
\eqref{j_lower} and~\eqref{j_upper} imply
\[
j_{\mu_n}(\tau_n;f_n)
= \mathcal{J}_{\mu_n}( \tau_n\varphi_1 + u_n^\top ; f_n )
\to \mathcal{J}_{\mu_0}( \tau_0\varphi_1 + w^\top ; f_0 )
= j_{\mu_0}(\tau_0;f_0) .
\]
Combining this result with
$\tau_n\to \tau_0$, $\mu_n\to \mu_0$,
$f_n\to f_0$ in $D^{-1,p'}(\mathbb{R}^N)$,
$u_n^\top \rightharpoonup w^\top$ weakly in $D^{1,p}(\mathbb{R}^N)$,
and
$u_n^\top \to w^\top$ strongly in $L^p(\mathbb{R}^N; m)$, we arrive at
\[
\left\| \tau_n\varphi_1 + u_n^\top \right\|_{ D^{1,p}(\mathbb{R}^N) }
\to
\left\| \tau_0\varphi_1 + w^\top \right\|_{ D^{1,p}(\mathbb{R}^N) }
\quad\mbox{as } n\to \infty .
\]
Thus, the uniform convexity of the standard norm on
$D^{1,p}(\mathbb{R}^N)$ forces
$\tau_n\varphi_1 + u_n^\top \to \tau_0\varphi_1 + w^\top$
strongly in $D^{1,p}(\mathbb{R}^N)$.
Our claim now follows as $\tau_n\to \tau_0$.
\endgroup
\end{remark}
%
Obviously, if the function
$j_{\lambda}\colon \mathbb{R}\to \mathbb{R}$ has a local minimum
at some point $\tau_0\in \mathbb{R}$, and
$u_0^\top$ is a global minimizer for the functional
$u^\top \mapsto \mathcal{J}_{\lambda}( \tau_0\varphi_1 + u^\top )$
on $D^{1,p}(\mathbb{R}^N)^\top$, then
$u_0 = \tau_0\varphi_1 + u_0^\top$ is a local minimizer for
$\mathcal{J}_{\lambda}$ on $D^{1,p}(\mathbb{R}^N)$ and thus
a weak solution to problem~\eqref{e:BVP.l}.
Our next lemma displays a similar result if
$j_{\lambda}$ has a {\it local maximum\/} at $\tau_0\in \mathbb{R}$;
it claims that
$\beta_{\lambda}$ in~\eqref{def.b_lam}
is a critical value of $\mathcal{J}_{\lambda}$.
%%%%% Criticality of the Maximin Point (Lemma)
\begin{lemma}\label{lem-Crit-Value}
Let\/
$0\leq \lambda\leq \Lambda_\infty - \eta$ and\/
$f\in D^{-1,p'}(\mathbb{R}^N)$.
Assume that the function
$j_{\lambda}\colon \mathbb{R}\to \mathbb{R}$ attains a local maximum
$\beta_{\lambda}$ at some point $\tau_0\in \mathbb{R}$.
Then there exists
$u_0^\top\in D^{1,p}(\mathbb{R}^N)^\top$ such that\/
$u_0^\top$ is a global minimizer for the functional\/
$u^\top \mapsto \mathcal{J}_{\lambda}( \tau_0\varphi_1 + u^\top )$
on $D^{1,p}(\mathbb{R}^N)^\top$,
$u_0 = \tau_0\varphi_1 + u_0^\top$ is a critical point for
$\mathcal{J}_{\lambda}$, and
$\mathcal{J}_{\lambda}(u_0) = \beta_{\lambda}$.
%
\end{lemma}
\begin{proof}
Given an arbitrary numerical sequence
$\{ \tau_n\}_{n=1}^\infty$ with
$\tau_n\to \tau_0$ in~$\mathbb{R}$ as $n\to \infty$ and
$\tau_n\not= \tau_0$ for all $n\geq 1$,
we can deduce from Remark~\ref{rem-Cont-Min}
that this sequence contains a subsequence
denoted again by $\{ \tau_n\}_{n=1}^\infty$, such that
for each $n=0,1,2,\dots$,
$u_n^\top$ is a global minimizer for the functional
$u^\top \mapsto \mathcal{J}_{\lambda}( \tau_n\varphi_1 + u^\top )$
and
$u_n^\top \to u_0^\top$ strongly in $D^{1,p}(\mathbb{R}^N)$
as $n\to \infty$.
It follows that
%
\begin{equation}
\begin{split}
\mathcal{J}_{\lambda}( \tau_n\varphi_1 + u_n^\top )
- \mathcal{J}_{\lambda}( \tau_0\varphi_1 + u_n^\top )
& \leq
\mathcal{J}_{\lambda}( \tau_n\varphi_1 + u_n^\top )
- \mathcal{J}_{\lambda}( \tau_0\varphi_1 + u_0^\top )\\
&= j_{\lambda}(\tau_n) - j_{\lambda}(\tau_0)
\leq 0
\end{split}
\label{c:J_upper}
\end{equation}
%
for all integers $n\geq 1$ sufficiently large;
again, we may assume it for all $n\geq 1$.
On the other hand, denoting
\[
\phi_n(s)\eqdef
\tau_0\varphi_1 + u_n^\top + s(\tau_n - \tau_0) \varphi_1
\quad\mbox{for }\ 0\leq s\leq 1 ;\ n\geq 1 ,
\]
we have
%
\begin{eqnarray*}
& \mathcal{J}_{\lambda}( \tau_n\varphi_1 + u_n^\top )
- \mathcal{J}_{\lambda}( \tau_0\varphi_1 + u_n^\top )
= (\tau_n - \tau_0) \int_0^1
\left\langle
\mathcal{J}_{\lambda}^\prime (\phi_n(s)) ,\, \varphi_1
\right\rangle \,{\rm d}s
\end{eqnarray*}
%
where
%
\begin{align*}
\left\langle
\mathcal{J}_{\lambda}^\prime (\phi_n(s)) ,\, \varphi_1
\right\rangle
& = \int_{\mathbb{R}^N}
|\nabla \phi_n(s)|^{p-2}\, \nabla \phi_n(s)
\cdot \nabla\varphi_1 \,{\rm d}x\\
& \quad
- \lambda \int_{\mathbb{R}^N}
|\phi_n(s)|^{p-2}\, \phi_n(s)\, \varphi_1\, m \,{\rm d}x
- \int_{\mathbb{R}^N} f\varphi_1 \,{\rm d}x .
\end{align*}
%
Since
$\phi_n(s)\to u_0 = \tau_0\varphi_1 + u_0^\top$
strongly in $D^{1,p}(\mathbb{R}^N)$ and
uniformly for $0\leq s\leq 1$, we arrive at
%
\begin{equation}
\begin{split}
& (\tau_n - \tau_0)^{-1}
\left[
\mathcal{J}_{\lambda}( \tau_n\varphi_1 + u_n^\top )
- \mathcal{J}_{\lambda}( \tau_0\varphi_1 + u_n^\top )
\right]
\\
& \longrightarrow\;
\left\langle
\mathcal{J}_{\lambda}^\prime (u_0) ,\, \varphi_1
\right\rangle
= \zeta_0\, \| \varphi_1\|_{ L^p(\mathbb{R}^N; m) } = \zeta_0
\quad\mbox{as } n\to \infty ,
\end{split}
\label{c:J_lower}
\end{equation}
%
where
%
\begin{align*}
\left\langle
\mathcal{J}_{\lambda}^\prime (u_0) ,\, \varphi_1
\right\rangle
& = \int_{\mathbb{R}^N}
|\nabla u_0|^{p-2}\, \nabla u_0
\cdot \nabla\varphi_1 \,{\rm d}x\\
& \quad
- \lambda \int_{\mathbb{R}^N} |u_0|^{p-2} u_0\varphi_1\, m \,{\rm d}x
- \int_{\mathbb{R}^N} f\varphi_1 \,{\rm d}x
\end{align*}
and
$\zeta_0\in \mathbb{R}$ is the Lagrange multiplier given by
$\mathcal{J}_{\lambda}^\prime (u_0) = \zeta_0\, m\, \varphi_1^{p-1}$.
Finally, if we choose $\tau_n$ such that
the sign of $(\tau_n - \tau_0)$
does not change for all $n=1,2,\dots$, then
\eqref{c:J_upper} and~\eqref{c:J_lower} yield
$\zeta_0\leq 0$ if $\mathop{\rm sgn}(\tau_n - \tau_0) = 1$, and
$\zeta_0\geq 0$ if $\mathop{\rm sgn}(\tau_n - \tau_0) = -1$.
Since both alternatives are possible, we conclude that
$\zeta_0 = 0$ which shows
$\mathcal{J}_{\lambda}^\prime (u_0) = 0$, i.e.,
$u_0$ is a weak solution to problem~\eqref{e:BVP.l_1} as desired.
In particular,
$\mathcal{J}_{\lambda}(u_0)$
is a critical value of $\mathcal{J}_{\lambda}$.
\end{proof}
%%%%% Criticality of the Maximin Point (Remark)
\begin{remark}\label{rem-Crit-Value}
\begingroup\rm
As an easy consequence of
\eqref{c:J_upper}, \eqref{c:J_lower}
in the proof of Lemma~\ref{lem-Crit-Value},
we conclude that the function
$j_{\lambda}\colon \mathbb{R}\to \mathbb{R}$ is differentiable at $\tau_0$ with
$j_{\lambda}^\prime (\tau_0) = 0$.
\endgroup
\end{remark}
%
\subsection{Rest of the proof of Theorem \ref{thm-Exist:p<2}}
\label{ss:Fredh_Mount}
We deduce from Lemma~\ref{lem-Saddle_Geom} that there exist
$a,b\in \mathbb{R}$ such that $a<0 0$ and
$\varrho\equiv \varrho(f^{\#}) > 0$ such that also with
$f = f^{\#} + \zeta\, m\varphi_1^{p-1}$
we have
%
\begin{eqnarray*}
& \max\{ j_{\lambda}(a;f) ,\, j_{\lambda}(b;f) \}
< j_{\lambda}(0;f)
\end{eqnarray*}
%
for all
$\lambda\in (\lambda_1 - \delta, \lambda_1 + \delta)$
and all
$\zeta\in (-\varrho,\varrho)$.
Now we can apply Lemma~\ref{lem-Crit-Value}
to conclude that the functional
$\mathcal{J}_{\lambda}(\,\cdot\, ;f)$
possesses a critical point
$u_1 = \tau_1\varphi_1 + u_1^\top$, with some
$\tau_1\in (a,b)$ and
$u_1^\top\in D^{1,p}(\mathbb{R}^N)^\top$.
This proves Theorem \ref{thm-Exist:p<2}.
\begin{proof}[Proof of Remark~\ref{rem-Exist:p<2}]
If $\lambda < \lambda_1$ then we have
$j_{\lambda}(\tau;f) \to +\infty$ as $|\tau|\to \infty$.
Consequently, for
$\lambda\in (\lambda_1 - \delta, \lambda_1)$
and
$\zeta\in (-\varrho,\varrho)$,
the continuous function
$j_{\lambda}(\,\cdot\, ;f) \colon \mathbb{R}\to \mathbb{R}$
possesses also a local minimizer in each of the intervals
$(-\infty, \tau_1)$ and $(\tau_1, \infty)$, say,
$\tau_2$ and $\tau_3$, respectively.
Our definition of $j_{\lambda}(\,\cdot\, ;f)$ now shows that
$u_2 = \tau_2\varphi_1 + u_2^\top$ and
$u_3 = \tau_3\varphi_1 + u_3^\top$
are local minimizers for
$\mathcal{J}_{\lambda}(\,\cdot\, ;f)$, with some
$u_2^\top, u_3^\top\in D^{1,p}(\mathbb{R}^N)^\top$, as claimed.
\end{proof}
%%%%% APPENDIX: ASYMPTOTICS OF THE FIRST EIGENFUNCTION
\section{Appendix: Asymptotics of the eigenfunction $\varphi_1$}
\label{s:Asymptotic}
To determine the asymptotic behavior of
the first eigenfunction $\varphi_1$ of
the $p$-Laplacian $\Delta_p$ on $\mathbb{R}^N$
subject to a weight $m(|x|)$, for $1 0} on the weight $m$ as follows:
%%%%%%%%%% MAIN HYPOTHESIS (weaker) %%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{enumerate}
\renewcommand{\labelenumi}{({\bf H'})}
%
\item
\makeatletter
\def\@currentlabel{{H'}} \label{hyp:m=0}
\makeatother
There exist constants $\delta > 0$ and $C>0$ such that
%
\begin{equation}
0\leq m(r)\leq \frac{C}{ (1+r)^{p+\delta} }
\quad\mbox{for almost all }\ 0\leq r < \infty ,
\label{ineq:m=0}
\end{equation}
%
and $m\not\equiv 0$ in $\mathbb{R}_+$.
\end{enumerate}
%
Under this hypothesis, we are able to establish
the following asymptotic behavior of
$u(r)$ and $u'(r)$ as $r\to \infty$.
%
%%%%% PROPOSITION: ASYMPTOTICS OF THE FIRST EIGENFUNCTION %%%%%%%
%
\begin{proposition}\label{prop-Asympt}
There exists a constant $c > 0$ such that
%
\begin{gather}
\lim_{r\to \infty}
\left( u(r)\, r^{ \frac{N-p}{p-1} } \right) = c ,
\label{e:u.infty} \\
\lim_{r\to \infty}
\left( u'(r)\, r^{ \frac{N-1}{p-1} } \right)
= - \textstyle\frac{N-p}{p-1}\, c .
\label{e:u'.infty}
\end{gather}
%
\end{proposition}
%
For the related Cauchy problem,
%
\begin{equation}
\label{ev:u:rel}
- \Delta_p u(|x|) = f(u(|x|)) \;\mbox{ for } x\in \mathbb{R}^N ;
\qquad
u(|x|)\to 0 \;\mbox{ as } |x|\to \infty ,
\end{equation}
%
with $f(u)\geq 0$ for $u>0$ sufficiently small,
the inequalities
$$
u(r)\, r^{ \frac{N-p}{p-1} } \geq c_1 > 0
\quad\mbox{and }\quad
- u'(r)\, r^{ \frac{N-1}{p-1} } \geq c_2 > 0
$$
for all sufficiently large $r>0$
(with some constants $c_1$ and $c_2$)
have been established in the work of
{\sc Ni} and {\sc Serrin} \cite[Theorem 6.1]{NiSerrin}.
Their method of proof applies also to our case.
%
For the inequality
%
\begin{equation}
\label{ev:u:ineq}
- \Delta_p u\leq m(|x|)\, u^{p-1} \quad\mbox{for } x\in \mathbb{R}^N ;
\quad
u(x)\to 0 \;\mbox{ as } |x|\to \infty ,
\end{equation}
%
with $u(x)$ not necessarily radially symmetric, $u(x) > 0$,
but with the weight $m(r)$ decaying at infinity faster than ours,
an upper estimate on the decay of $u$ at infinity can be found in
{\sc Fleckinger}, {\sc Harrell} and {\sc de Th\'elin}
\cite[Theorem IV.2]{FHdT}.
In the proof of Proposition~\ref{prop-Asympt}
we need a few auxiliary results.
The Cauchy problem \eqref{ev:u} is equivalent to
%
\begin{equation}
\label{ev:u.rad}
\begin{gathered}
{}- (|u'|^{p-2} u')' - \frac{N-1}{r}\, |u'|^{p-2} u'
= m(r)\, u^{p-1} \quad\mbox{for } r>0 ;
\\
u'(r)\to 0 \mbox{ as } r\to 0 \quad\mbox{and }\quad
u(r) \to 0 \mbox{ as } r\to \infty .
\end{gathered}
\end{equation}
%
This problem can be rewritten as
%
\begin{equation}
\label{ev:r.u.rad}
\begin{gathered}
{}- ( r^{N-1}\, |u'|^{p-2} u' )' = m(r)\, r^{N-1}\, u^{p-1}
\quad\mbox{for } r>0 ; \\
u'(r)\to 0 \mbox{ as } r\to 0 \quad\mbox{and }\quad
u(r) \to 0 \mbox{ as } r\to \infty .
\end{gathered}
\end{equation}
%
We reduce this second-order differential equation to
a first-order equation by introducing
the Riccati-type transformation
%
\begin{equation}
\label{u:Riccati}
U(r)\eqdef
{}- r^{p-1}
\Big| \frac{ u'(r) }{ u(r) } \Big|^{p-2}
\frac{ u'(r) }{ u(r) }
\quad\mbox{for } r>0 ,\quad
U(0)\eqdef 0 .
\end{equation}
%
By~\eqref{ev:r.u.rad}, the function
$r\mapsto r^{ \frac{N-1}{p-1} } u'(r)$
is nonincreasing for $0 < r < \infty$ which implies
$u'(r)\leq 0$ for all $r>0$, and therefore also
$U(r)\geq 0$.
Hence, for $r>0$,
%
\begin{align*}
U'(r)
& =
- (p-1) r^{p-2}
\big| \frac{u'}{u} \big|^{p-2} \frac{u'}{u}
- (p-1) r^{p-1}
\big| \frac{u'}{u} \big|^{p-2}
\big[ \frac{u''}{u} - \big(\frac{u'}{u}\big)^2 \big]
\\
& = \frac{p-1}{r}\, U(r)
- r^{p-1}\, \frac{ ( |u'|^{p-2} u' )' }{ |u|^{p-2} u }
+ \frac{p-1}{r}\, U(r)^{ \frac{p}{p-1} } .
\end{align*}
%
Inserting the second derivative expression from
equation \eqref{ev:u.rad}, we arrive at
\[
U'(r) =
- \frac{N-p}{r}\, U(r)
+ \frac{p-1}{r}\, U(r)^{ \frac{p}{p-1} }
+ m(r)\, r^{p-1} .
\]
This is a differential equation for the unknown function $U$
which we rewrite as
%
\begin{equation}
\label{eq:U}
U'(r)
= \frac{p-1}{r}\, U(r)
\Big( U(r)^{ \frac{1}{p-1} } - \frac{N-p}{p-1} \Big)
+ m(r)\, r^{p-1}
\quad\mbox{for } r>0 .
\end{equation}
%
An upper bound for $U(r)$ is obtained first:
%%%%% LEMMA - Upper Bound for U(r)
\begin{lemma}\label{lem-UpperBound}
We have
%
\begin{equation}
\label{e:U.Upper}
U(r)\leq c_{N,p}\eqdef \big(\frac{N-p}{p-1}\big)^{p-1}
\quad\mbox{for all } r\geq 0 .
\end{equation}
\end{lemma}
\begin{proof}
Clearly, by \eqref{u:Riccati}, the function
$U\colon \mathbb{R}_+\to \mathbb{R}$ is continuous and, by \eqref{eq:U},
it is differentiable almost everywhere with the derivative
$U'$ being locally bounded.
Now, in contradiction with \eqref{e:U.Upper},
suppose that there exists a number $r_0\geq 0$ such that
$U(r_0) > c_{N,p}$.
Let
$$
r_1\eqdef
\sup\{ r'\colon r'\geq r_0 \,\mbox{ and }\, U(r) > c_{N,p}
\;\mbox{ for all }\; r_0\leq r\leq r' \} .
$$
Next we show that $r_1 = \infty$.
Indeed, equation \eqref{eq:U} with
$U(r) > c_{N,p}$ and $m(r)\geq 0$ for $r_0\leq r < r_1$ implies
$U'(r) > 0$.
This shows that the function $U(r)$ is strictly increasing for
$r_0\leq r < r_1$.
Consequently, $r_1 < \infty$ would yield
$U(r_1) = c_{N,p} < U(r_0) < U(r_1)$ which is impossible.
Hence, there is a constant $\gamma > 0$ such that
the expression inside the parenthesis in eq.~\eqref{eq:U}
satisfies
%
\[
U(r)^{ \frac{1}{p-1} } - \frac{N-p}{p-1}
\geq \gamma\, U(r)^{ \frac{1}{p-1} }
\quad\mbox{for all } r\geq r_0 .
\]
%
Applying this inequality to equation \eqref{eq:U}
we obtain
%
\[
U'(r)\geq \frac{p-1}{r}\, \gamma\, U(r)^{ \frac{p}{p-1} }
\quad\mbox{for all } r\geq r_0 .
\]
%
We integrate this inequality over the interval
$[r_0,r]$ to get
$$
U(r_0)^{ - \frac{1}{p-1} } - U(r)^{ - \frac{1}{p-1} }
\geq \gamma\, \log (r/r_0)
\quad\mbox{for all } r\geq r_0 .
$$
Recalling $U(r) > 0$ and letting $r\to \infty$,
we arrive at
$U(r_0)^{ - \frac{1}{p-1} } \geq +\infty$,
which is a contradiction.
Inequality \eqref{e:U.Upper} is proved.
\end{proof}
Define the function
%
\begin{equation}
\label{def:a(r)}
a(r)\eqdef
\frac{p-1}{r}
\Big( \frac{N-p}{p-1} - U(r)^{ \frac{1}{p-1} } \Big)
\quad\mbox{for } r>0 .
\end{equation}
%
Note that $a(r)\geq 0$ by Lemma~\ref{lem-UpperBound},
and
$$
a(r) = \frac{N-p}{r} + (p-1) \,\frac{u'(r)}{u(r)}
= (p-1) \,\frac{ {\rm d} }{ {\rm d}r }
\log\left( u(r)\, r^{ \frac{N-p}{p-1} } \right) .
$$
We substitute this function into eq.~\eqref{eq:U}
and use integrating factor to integrate it over
any interval $[r_0,r]$ with $r_0 > 0$ fixed and $r\geq r_0$.
We thus obtain
%
\begin{equation}
\label{eq:U(r)}
U(r) - U(r_0)\, e^{ - \int_{r_0}^r a(s) \,{\rm d}s }
= \int_{r_0}^r m(s)\, s^{p-1}\,
e^{ - \int_s^r a(t) \,{\rm d}t } \,{\rm d}s .
\end{equation}
%
Furthermore, we introduce the abbreviation
%
\begin{equation}
\label{def:A(r)}
A(r)\eqdef \int_{r_0}^r a(s) \,{\rm d}s
= (p-1)\, \log
\frac{ u(r)\, r^{ \frac{N-p}{p-1} } }
{ u(r_0)\, r_0^{ \frac{N-p}{p-1} } }
\quad\mbox{for } r\geq r_0 .
\end{equation}
%
%%%%% LEMMA - Lower Bound for U(r)
\begin{lemma}\label{lem-LowerBound}
We have $a(r)\geq 0$ for all $r>0$ and
%
\begin{equation}
\label{e:U.Lower}
\int_{r_0}^\infty a(r) \,{\rm d}r < \infty
\quad\mbox{for every } r_0 > 0 .
\end{equation}
%
\end{lemma}
%
\begin{proof}
The function $A(r)$ is nondecreasing for $r_0\leq r < \infty$.
Now, suppose that
$\lim_{r\to \infty} A(r) = +\infty$.
From equation \eqref{eq:U(r)} we deduce
%
\begin{equation}
\label{ineq:U(r)}
0 \leq U(r) - U(r_0)\, e^{- A(r)} \leq
\int_{r_0}^\infty m(s)\, s^{p-1}\,
e^{ - ( A(r) - A(s) ) } \,{\rm d}s
\quad\mbox{for } r\geq r_0 .
\end{equation}
%
Due to our hypothesis \eqref{hyp:m=0}, we are allowed to apply
Lebesgue's dominated convergence theorem to the last integral
to obtain, as $r\to \infty$,
$0\leq \lim_{r\to \infty} U(r)\leq 0$, i.e.,
$\lim_{r\to \infty} U(r) = 0$.
This shows that, given any number $\eta$ such that
$0 < \eta < N-p$,
there exists a number $r_\eta\geq r_0$ such that
$$
a(r) =
\frac{p-1}{r}
\Big( \frac{N-p}{p-1} - U(r)^{ \frac{1}{p-1} } \Big)
\geq \frac{N - p - \eta}{r}
\quad\mbox{for all } r\geq r_\eta .
$$
Since $r_0$ is arbitrary, $r_0 > 0$,
we may take $r_0 = r_\eta$.
Upon integration, we get
%
\begin{equation}
\label{ineq:A(r)}
A(r)\geq (N - p - \eta) \int_{r_0}^r \frac{ {\rm d}s }{s}
= \log \left( (r/r_0)^{N - p - \eta} \right)
\quad\mbox{for all } r\geq r_0 .
\end{equation}
%
We apply inequalities
\eqref{ineq:m=0} and \eqref{ineq:A(r)} to equation \eqref{eq:U(r)}
to obtain for all $r\geq 0$,
%
\begin{equation}
\label{est:|U|}
\begin{split}
U(r)&\leq
U(r_0) \big(\frac{r}{r_0}\big)^{ - (N - p - \eta) }
+ C \int_{r_0}^r \frac{ s^{p-1} }{ (1+s)^{p+\delta} }
\big(\frac{r}{s}\big)^{ - (N - p - \eta) }
\,{\rm d}s \\
& \leq
U(r_0) \big(\frac{r}{r_0}\big)^{ - (N - p - \eta) }
+ \frac{ C\, r^{ - (N - p - \eta) } }{ N - p - \eta - \delta }
\left(
r^{ N - p - \eta - \delta }
- r_0^{ N - p - \eta - \delta }
\right) .
\end{split}
\end{equation}
%
Note that in inequality~\eqref{ineq:m=0},
the constant $\delta > 0$ may be chosen arbitrarily small;
we choose it such that
$0 < \delta < N - p - \eta$.
Hence, \eqref{est:|U|} yields
$$
U(r)\leq C_0\, r^{-\delta}
\quad\mbox{for all } r\geq r_0 ,
$$
where $C_0 > 0$ is a constant.
With our definition of $U$ we have equivalently
$$
- \frac{u'(r)}{u(r)} \leq
C_0^{ \frac{1}{p-1} }\, r^{ - 1 - \frac{\delta}{p-1} }
\quad\mbox{for all } r\geq r_0 .
$$
Upon integration we get
$$
{}- \log \frac{u(r)}{u(r_0)}
\leq C_0'
\big(
r_0^{ - \frac{\delta}{p-1} }
- r^{ - \frac{\delta}{p-1} }
\big)
\quad\mbox{for all } r\geq r_0 ,
$$
where $C_0'> 0$ is a constant.
Recalling $u(r)\to 0$ as $r\to \infty$,
we arrive at
$+\infty\leq C_0'\, r_0^{ - \frac{\delta}{p-1} }$
which is absurd.
The proof of the lemma is complete.
\end{proof}
Finally, we determine the limit of the function $U$ at infinity.
%%%%% LEMMA - Limit of U(r)
\begin{lemma}\label{lem-U.Limit}
We have
%
\begin{equation}
\label{e:U.Limit}
\lim_{r\to \infty} U(r) = c_{N,p}
= \big(\frac{N-p}{p-1}\big)^{p-1} .
\end{equation}
\end{lemma}
%
\begin{proof}
The limit
\[
A(\infty)\eqdef \lim_{r\to \infty} A(r)
= \int_{r_0}^\infty a(r) \,{\rm d}r
\]
exists and satisfies $0\leq A(\infty) < \infty$,
by \eqref{def:A(r)} and~\eqref{e:U.Lower}.
We apply this fact and hypothesis \eqref{hyp:m=0}
to equation \eqref{eq:U(r)} to obtain the existence of the limit
%
\begin{equation}
\label{lim:U(r)}
U(\infty)\eqdef \lim_{r\to \infty} U(r)
= U(r_0)\, e^{- A(\infty)}
+ \int_{r_0}^\infty m(s)\, s^{p-1}\,
e^{ - ( A(\infty) - A(s) ) } \,{\rm d}s ,
\end{equation}
%
using Lebesgue's dominated convergence theorem.
We have $U(\infty)\leq c_{N,p}$ by~\eqref{e:U.Upper}.
However, if $U(\infty) < c_{N,p}$ then
there exist constants $\gamma > 0$ and $r_1\geq r_0$ such that
$$
a(r) =
\frac{p-1}{r}
\Big( \frac{N-p}{p-1} - U(r)^{ \frac{1}{p-1} } \Big)
\geq \frac{\gamma}{r}
\quad\mbox{for all } r\geq r_1 .
$$
But this inequality contradicts~\eqref{e:U.Lower}.
We have proved~\eqref{e:U.Limit}.
\end{proof}
Finally, we are ready to derive formulas
\eqref{e:u.infty} and~\eqref{e:u'.infty}.
\begin{proof}[Proof of Proposition~\ref{prop-Asympt}]
We combine \eqref{def:A(r)} and \eqref{e:U.Lower}
to conclude that the limit
$$
c_0\eqdef \lim_{r\to \infty} \log
\Big(
{ u(r)\, r^{ \frac{N-p}{p-1} } } \Big\slash
{ u(r_0)\, r_0^{ \frac{N-p}{p-1} } }
\Big)
$$
exists and satisfies $0\leq c_0 < \infty$.
The desired formula \eqref{e:u.infty}
follows immediately with
$c\eqdef e^{c_0}\, u(r_0)\, r_0^{ \frac{N-p}{p-1} } > 0$.
%
The convergence formula \eqref{e:U.Limit} reads
%
\begin{equation}
\label{e:u'/u.infty}
{}- r \,\frac{u'(r)}{u(r)} \,\longrightarrow\,
\frac{N-p}{p-1}
\quad\mbox{as } r\to \infty .
\end{equation}
%
We combine this result with \eqref{e:u.infty}
to get~\eqref{e:u'.infty}.
%
The proposition is proved.
\end{proof}
\subsection*{Acknowledgment}
This work was supported in part by
le Minist\`ere des Affaires \'Etrang\`eres (France)
and
the German Academic Exchange Service (DAAD, Germany)
within the exchange program ``PROCOPE''.
A part of this research was performed when
P.~T.\ was a visiting professor at CEREMATH,
Universit\'e Toulouse~1 -- Sciences Sociales,
Toulouse, France.
A part of the research reported here was performed when
Peter Tak\'a\v{c} was a visiting professor at CEREMATH,
Universit\'e Toulouse~1 -- Sciences Sociales, Toulouse, France.
The authors express their thanks to an anonymous referee for
careful reading of the manuscript and a couple of pertinent references.
\begin{thebibliography}{00}
\bibitem{Anane-1}
A. Anane,
{\it Simplicit\'e et isolation de la premi\`ere valeur
propre du $p$-laplacien avec poids},
Comptes Rendus Acad. Sc. Paris, S\'erie I,
{\bf 305} (1987), 725--728.
\bibitem{BergSchecht}
M.~S. Berger and M.~Schechter,
{\it Embedding theorems and quasilinear elliptic boundary value
problems for unbounded domains},
Trans. Amer. Math. Soc., {\bf 172} (1972), 261--278.
\bibitem{Drabek-1}
P. Dr\'abek,
{\sl ``Solvability and Bifurcations of Nonlinear Equations''},
Pitman Research Notes in Mathematics Series,
Vol.~{\bf 264}.
Longman Scientific {\&} Technical, U.K., 1992.
\bibitem{DrabGirgMan}
P. Dr\'abek, P. Girg, and R.~F. Man\'asevich,
{\it Generic Fredholm alternative\--type results for
the one\--dimensional $p$-Laplacian},
Nonlinear Diff. Equations and Appl., {\bf 8} (2001), 285--298.
\bibitem{DrabHolub}
P. Dr\'abek and G. Holubov\'a,
{\it Fredholm alternative for the $p$-Laplacian in higher dimensions},
J. Math. Anal. Appl., {\bf 263} (2001), 182--194.
\bibitem{FGdT}
J. Fleckinger, J.-P. Gossez, and F. de~Th\'elin,
{\it Antimaximum principle in $\mathbb{R}^N$: local versus global},
J. Differential Equations, {\bf 196} (2004), 119--133.
\bibitem{FHdT}
J.~Fleckinger, E.~M. Harrell and F.~de Th\'elin,
{\it Asymptotic behavior of solutions for some nonlinear
partial differential equations on unbounded domains},
Electr. J.~Diff. Equations {\bf 77} (2001), 1--14.
\bibitem{FMST}
J. Fleckinger, R.~F. Man\'asevich, N.~M. Stavrakakis, and
F. de~Th\'elin,
{\it Principal eigenvalues for some quasilinear elliptic equations
in $\mathbb{R}^N$},
Advances Diff. Equations, {\bf 2} (1997), 981--1003.
\bibitem{FleckTakac-1}
J. Fleckinger and P. Tak\'a\v{c},
{\it An improved Poincar\'e inequality and
the $p$-Laplacian at resonance for $p>2$},
Advances Diff. Equations, {\bf 7}(8) (2002), 951--971.
\bibitem{FucikNSS}
S. Fu\v{c}\'{\i}k, J. Ne\v{c}as, J. Sou\v{c}ek, and V. Sou\v{c}ek,
{\sl ``Spectral Analysis of Nonlinear Operators''},
Lecture Notes in Mathematics, Vol.~{\bf 346}.
Springer\--Verlag, New York\--Berlin\--Hei\-del\-berg, 1973.
\bibitem{GilbargTrud}
D.~Gilbarg and N.~S. Trudinger,
{\sl ``Elliptic Partial Differential Equations of Second Order''},
Springer-Verlag, New York--Berlin--Heidelberg, 1977.
\bibitem{Kato}
T. Kato,
{\sl ``Perturbation Theory for Linear Operators''},
in {\em Grundlehren der ma\-the\-ma\-ti\-schen Wissenschaften},
Vol.~{\bf 132}.
Springer\--Verlag, New York\--Berlin\--Hei\-del\-berg, 1980.
\bibitem{Kufner}
A. Kufner,
{\sl ``Weighted Sobolev Spaces''},
John Wiley {\&} Sons Ltd., New York, 1985.
\bibitem{Lindqvist}
P. Lindqvist,
{\it On the equation
$\mathrm{div}(\vert\nabla u\vert^{p-2} \nabla u)$ $+$
$\lambda \vert u\vert^{p-2} u$ $= 0$},
Proc. Amer. Math. Soc., {\bf 109}(1) (1990), 157--164.
\bibitem{ManTakac}
R.~F. Man\'asevich and P. Tak\'a\v{c},
{\it On the Fredholm alternative for
the $p$-Laplacian in one dimension},
Proc. London Math. Soc., {\bf 84} (2002), 324--342.
\bibitem{NiSerrin}
W.-M. Ni and J.~B. Serrin,
{\it Existence and nonexistence theorems for ground states of
quasilinear partial differential equations.
The anomalous case.}
Rome, Accad. Naz. dei Lincei, Atti dei Convegni
{\bf 77} (1986), 231--257.
\bibitem{PinoDrabMan}
M.~A. del~Pino, P. Dr\'abek, and R.~F. Man\'asevich,
{\it The Fredholm alternative at the first eigenvalue for
the one\--dimensional $p$-Laplacian},
J. Differential Equations, {\bf 151} (1999), 386--419.
\bibitem{Rabin}
P.~H. Rabinowitz,
{\sl ``Minimax Methods in Critical Point Theory with
Applications to Differential Equations''},
CBMS Series in Mathematics, Vol.~{\bf 65},
Amer. Math. Soc., Providence, R.I., 1986.
\bibitem{Schecht-1}
M. Schechter,
{\it On the essential spectrum of an elliptic operator
perturbed by a potential},
J. Analyse Math., {\bf 22} (1969), 87--115.
\bibitem{Schecht-2}
M. Schechter,
{\sl ``Spectra of Partial Differential Operators''},
North\--Holland Publ. Co., New York\--Amsterdam\--Oxford, 1971.
\bibitem{StavrThelin}
N.~M. Stavrakakis and F. de~Th\'elin,
{\it Principal eigenvalues and antimaximum principle for
some quasilinear elliptic equations in $\mathbb{R}^N$},
Math. Nachrichten, {\bf 212} (2000), 155--171.
\bibitem{Takac-1}
P. Tak\'a\v{c},
{\it On the Fredholm alternative for
the $p$-Laplacian at the first eigenvalue},
Indiana Univ. Math. J., {\bf 51}(1) (2002), 187--237.
\bibitem{Takac-2}
P. Tak\'a\v{c},
{\it On the number and structure of solutions for
a Fredholm alternative with the $p$-Laplacian},
J. Differential Equations, {\bf 185} (2002), 306--347.
\bibitem{Tolksdorf-1}
P. Tolksdorf,
{\it On the Dirichlet problem for quasilinear equations
in domains with conical boundary points},
Comm. P.D.E., {\bf 8}(7) (1983), 773--817.
\bibitem{Vazquez}
J.~L. V\'azquez,
{\it A strong maximum principle for
some quasilinear elliptic equations},
Appl. Math. Optim., {\bf 12} (1984), 191--202.
\bibitem{Yosida}
K. Yosida,
{\sl ``Functional Analysis''}, 6th Ed.,
in {\em Grundlehren der ma\-the\-ma\-ti\-schen Wissenschaften},
Vol.~{\bf 123}.
Springer\--Verlag, New York\--Berlin\--Hei\-del\-berg, 1980.
\end{thebibliography}
\end{document}