\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Elliptic equations with one-sided critical growth \hfil EJDE--2002/89} {EJDE--2002/89\hfil Marta Calanchi \& Bernhard Ruf \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 89, pp. 1--21. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Elliptic equations with one-sided critical growth % \thanks{ {\em Mathematics Subject Classifications:} 35J20. \hfil\break\indent {\em Key words:} Nonlinear elliptic equation, critical growth, linking structure. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted March 01, 2002. Published October 18, 2002.} } \date{} % \author{Marta Calanchi \& Bernhard Ruf} \maketitle \begin{abstract} We consider elliptic equations in bounded domains $\Omega\subset \mathbb{R}^N$ with nonlinearities which have critical growth at $+\infty$ and linear growth $\lambda$ at $-\infty$, with $\lambda > \lambda_1$, the first eigenvalue of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms provided $N \ge 6$. In dimensions $N = 3,4,5$ an additional lower order growth term has to be added to the nonlinearity, similarly as in the famous result of Brezis-Nirenberg for equations with critical growth. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \numberwithin{equation}{section} \section{Introduction} We consider the superlinear problem $$\label{p1} \begin{gathered} -\Delta u = \lambda u +u_+^{2^\star -1} +g(x,u_+)+f(x) \quad \mbox{in } \Omega \\ u= 0 \quad \mbox {on } \partial \Omega \end{gathered}$$ where $\Omega \subset \mathbb{R}^N\; (N\ge 3)$ is a bounded domain with smooth boundary, $2^\star =2N/(N-2)$ is the critical Sobolev exponent, $g(\cdot,s_+)\in C (\bar{\Omega}\times \mathbb{R}, \mathbb{R}^+)$ has a subcritical growth at infinity, and $s_+=\max\{s,0\}$. Furthermore, we assume that $\lambda > \lambda_1$, the first eigenvalue of the Laplacian. This means that the function $$k(s) = \lambda s + s_+^{2^\star -1} + g(x,s_+)$$ interferes" with all but a finite number of eigenvalues of the Laplacian, in the sense that $$\lambda_1 < \lambda = \lim_{s \to - \infty} \frac{k(s)}{s} < \lim_{s \to +\infty} \frac{k(s)}{s} = + \infty$$ For subcritical nonlinearities, such problems have been treated by Ruf-Srikanth \cite{RS} and de Figueiredo \cite{D}, proving the existence of at least two solutions provided that the forcing term $f(x)$ has the form $$\label{eq1} f(x) = h(x) + t e_1(x),$$ where $h \in L^r({\Omega})$, for some $r>N$, is given, $e_1$ is the (positive and normalized) first eigenfunction of the Laplacian, and $t > T$, for some sufficiently large number $T=T(h)$. We remark that the search of solutions for forcing terms of the form (\ref{eq1}) is natural if the nonlinearity crosses {\it all} eigenvalues, i.e. $\lambda<\lambda_1$ (then the problem is of so-called Ambrosetti-Prodi type); indeed, in this case there exists an obvious {\it necessary condition} of the form $\int_\Omega f e_1 dx \lambda_1$\\ ii) $h\in L^r(\Omega), r > N$, is given, with $h\in \ker(-\Delta-\lambda)^\perp$ if $\lambda$ is an eigenvalue.\\ Then there exists $T_0=T_0(h)>0$ such that if $t>T_0$ then problem (\ref{p2a}) has a negative solution $\phi_t\in W^{2,r}\cap W^{1,r}_0\subset C^{1,1-N/r}$. Suppose in addition that\\ iii) $\lambda$ is {\rm not} an eigenvalue of $(-\Delta,H_0^1(\Omega))$ \\ iv) $N \ge 7$ \\ Then problem (\ref{p2a}) has a second solution for $t>T_0$. \end{theorem} We remark that in \cite{DJ} only $h\in L^2(\Omega)$ is assumed, however this seems not sufficient to get the stated results. In this paper we improve and extend the result of de Figueiredo and Jianfu in the following ways: \paragraph{Main Results:} \begin{description} \item[$1)$] $\lambda > \lambda_1$ and $\lambda$ is not an eigenvalue:\\ $N \ge 6$: if $h \in L^r(\Omega)$ ($r > N$), then equation (\ref{p1}) has a second solution for $f = h + te_1$, with $t > T_0(h)$. \\ $N = 3, 4, 5$: if $h \in L^r(\Omega)$ ($r > N$) and $g(x,s_+) \ge c s^q$ for some $q=q(N)>1$ and $c>0$, then equation (\ref{p1}) has a second solution for $f = h + te_1$, with $t > T_0$. \item[$2)$] $\lambda$ is an eigenvalue, i.e. $\lambda = \lambda_k$, for some $k \ge 2$: If $h \in L^r(\Omega)$ ($r > N$) satisfies $h \perp ker (-\Delta-\lambda_k, H_0^1(\Omega))$, then equation (\ref{p1}) has a second solution for $f = h + te_1$ with $t > T_0$ (in dimensions $N = 3,4,5$ the assumptions made in 1) have to be added). \item[$3)$] $\lambda$ in a left neighborhood of $\lambda_k$, $k\ge 2$: There exists a $\delta > 0$ such that if $\lambda \in (\lambda_k - \delta, \lambda_k)$ and $h \in L^r(\Omega)$ ($r > N$) satisfies $h \perp ker (-\Delta, H_0^1(\Omega)$, then equation (\ref{p1}) has at least three solutions for $f = h + te_1$, with $t > T_0$ (in dimensions $N = 3,4,5$ the assumptions made in 1) have to be added). \end{description} For proving these statements, one proceeds as follows: the first (negative) solution is easily obtained (see \cite{RS}, \cite{D}, \cite{DJ}). To obtain a second solution, one uses the saddle point structure around the first solution and applies the generalized mountain pass theorem of Rabinowitz \cite{Ra}. To prove the Palais-Smale condition, one proceeds as in \cite{BN}, using a sequence of concentrating functions (obtained from the so-called Talenti function). However due to the presence of the first solution, lower order terms appear in the estimates. To handle these estimates, an orthogonalization" procedure based on separating the supports of the Talenti sequence and the (approximate) first solution is used (this approach was introduced in \cite{GR}). With this method we obtain the results 1) and 2). To prove 3), one shows that the branch" of solutions with $\lambda \in (\lambda_k,\lambda_{k+1})$ can be extended to $\lambda = \lambda_k$ if $h \perp ker(-\Delta)$. Actually, this branch can be extended slightly beyond $\lambda_k$, i.e. to $\lambda \in (\lambda_k - \delta, \lambda_k]$, and the corresponding solutions are clearly bounded away from the negative solution. Since in $\lambda_k$ starts a bifurcation branch $(\lambda,u)$ emanating from the negative solution and bending to the left (as shown by de Figueiredo - Jianfu \cite{DJ}), we conclude that for $\lambda$ to the left of and close to $\lambda_k$ there exist at least three solutions. \section{Statement of theorems} In this section we give the precise statements of the theorems. Furthermore, the notation and basic properties are introduced. We consider problem (\ref{p1}) under the following conditions on the nonlinearity $g$: \begin{description} \item[$(g_1)$] $g:\bar{\Omega}\times\mathbb{R}\rightarrow \mathbb{R}^+$ is continuous; \item[$(g_2)$] $g(x,s) \equiv 0$ for $s \le 0$, i.e. $g(x,s) = g(x,s_+)$ with $g(x,0) = 0$; \item[$(g_3)$] There exist constants $c_1>0$ and $10$ and $q$ with $\max \left\{ \frac{N}{2(N-1)} \; ,\; \frac{2}{3} \right\} < \frac{q+1}{2^\star} <1$, such that $g(x,s_+)\ge c(s_+)^{q}$, for all $s \in \mathbb R$. \end{description} We prove the following results: \begin{theorem}\label{t1} $N\ge 6$: Let $h \in L^r(\Omega), r > N$, be given, and let $T_0=T_0(h)$ the number given by Theorem 0. Assume that $\lambda > \lambda_1$, and that $g$ satisfies $(g_1), (g_2), (g_3)$. If $\lambda$ is an eigenvalue, say $\lambda = \lambda_k$, assume in addition that $h \perp ker(-\Delta -\lambda_k)$. Then problem (\ref{p1}) has a second solution for $f = h + te_1$, with $t > T_0$. \end{theorem} \begin{theorem} \label{t2} $3 \le N \le 5$: Let $h \in L^r, r > N$, be given, and suppose that all other assumptions of Theorem \ref{t1} are satisfied. If $g$ satisfies also $(g_4)$, then problem (\ref{p1}) has a second solution for $f = h + te_1$, with $t > T_0$. \end{theorem} \begin{theorem} Assume the hypotheses of Theorems \ref{t1} and \ref{t2}, and assume in addition that $h \perp \ker(-\Delta-\lambda_k)$, for some $k\ge 2$. Then there exists $\delta > 0$ such that for $\lambda \in (\lambda_k-\delta,\lambda_k)$ problem (\ref{p1}) has at least three solutions for $f = h + te_1$, with $t > T_0$. \end{theorem} The first solution of equation (\ref{p1}) is a negative solution, and its existence, for $t$ sufficiently large, is not difficult to prove (see \cite{D}, \cite{DJ}): first note that a negative solution satisfies the {\it linear equation} $$\label{p2} \begin{gathered} -\Delta y - \lambda y = h + t e_1 \quad \mbox{in } \Omega\\ y = 0 \quad \mbox {on } \partial \Omega \end{gathered}$$ The solution of this equation is unique, and we denote it, in dependence of $te_1$, by $\phi_t$ (we remark that if $\lambda = \lambda_k$ then $h \perp ker(-\Delta - \lambda_k)$ is required, and the solution is unique in $(ker (-\Delta - \lambda_k))^\perp$). Note that we may assume that $\int_\Omega h e_1 dx = 0$, and then the solution $\phi_t$ can be written as $\phi_t = w + s_t e_1$, with $\int_\Omega w e_1 dx = 0$ and $s_t = t/(\lambda_1 -\lambda)$ (and with $w \perp ker (-\Delta - \lambda_k)$ if $\lambda = \lambda_k$). Since $h \in L^r, r > N$, we have $w \in C^{1,1-N/r}({\overline \Omega})$, and it is known (see \cite{GT}) that on $\partial \Omega$ the (interior) normal derivative $\frac{\partial}{\partial n} e_1(x)$ is positive; hence $\phi_t < 0$ for $t$ sufficiently large. To find a second solution of equation (\ref{p1}), we set $u=v+\phi_t$; then $v$ solves $$\label{p3} \begin{gathered} -\Delta v = \lambda v +(v+\phi_t )_+^{2^\star-1} +g(x,(v+\phi_t)_+) \quad \mbox {in } \Omega \\ v = 0 \quad \mbox{on } \partial \Omega \end{gathered}$$ Clearly $v = 0$ is a solution of this equation, corresponding to the negative solution $\phi_t$ for equation (\ref{p1}). To find a second solution of equation (\ref{p3}) one can look for non trivial critical points of the functional $J(v) = \frac{1}{2}\int_\Omega ( |\nabla v|^2-\lambda v^2)dx -\frac{1}{2^\star}\int_\Omega (v+\phi_t)_+^{2^\star}dx -\int_\Omega G(x, (v+\phi_t)_+ )dx,$ where $G(x,s) : = \int_0^s g(x,\xi )\,d\xi$. This was the approach of de Figueiredo-Jianfu in \cite{DJ}. For applying the Generalized Mountain Pass theorem of Rabinowitz \cite{Ra} one needs to prove some geometric estimates. Furthermore, since the nonlinearities have critical growth, one needs to show that the minimax level avoids the non-compactness levels given by the concentrating sequences" $u_\epsilon$ (see \cite{BN} and below). In these estimates, the terms of the form $v + \phi_t + u_\epsilon$ are not easy to handle. In this paper we apply a method introduced in \cite{GR} to make such estimates easier. The idea consists in separating the supports of $v + \phi_t$ and $u_\epsilon$ by concentrating the support of the functions $u_\epsilon$ in small balls, and cutting small holes" into the functions $v + \phi_t$ such that the respective supports are disjoint. These manipulations create some errors, but these are easier to handle than to estimate the mixed terms" arising in expressions like $(v + \phi_t + u_\epsilon)_+^p$. Moving these small holes'' near $\partial \Omega$ where the first solution $\phi_t$ is small allows to further improve the estimates. \section{Variational setting and preliminary properties} We begin by replacing equation (\ref{p3}) by an approximate equation. We denote by $B_r(x_0) \subset \Omega$ a ball of radius $r$ and center $x_0 \in \Omega$. Choose $m \in \mathbb{N}$ so large that $B_{2/m}(x_0) \subseteq \Omega$, and let $\eta_m \in C_0^\infty(\Omega)$ such that $0 \le \eta_m(x) \le 1$, $|\nabla \eta_m (x)| \le 2m$ and $\eta_m(x)= \begin{cases} 0 & \mbox {in } B_{1/m}(x_0)\\ 1 & \mbox {in } \Omega \setminus B_{2/m}(x_0) \end{cases}$ Define the functions $\phi_t^m=\eta_m\phi_t$ and set $f_m = -\Delta\phi_t^m-\lambda\phi_t^m.$ Setting as before $u=v+\phi_t^m$ in equation (\ref{p1}), we see that then $v = \phi_t-\phi_t^m$ solves the equation $$\label{p4} \begin{gathered} -\Delta v = \lambda v +(v+\phi_t^m)_+^{2^\star-1} +g(x,(v+\phi_t^m)_+) +( f-f_m ) \mbox{ in } \Omega \\ v= 0 \quad \mbox{on } \partial \Omega \end{gathered}$$ Clearly $v=\phi_t-\phi_t^m$ corresponds to the trivial solution of this equation; for finding other solutions of (\ref{p4}) we look for critical points of the functional, $J:H \to \mathbb{R}$, \begin{align*} J(v)=&\frac{1}{2}\int_\Omega ( |\nabla v|^2-\lambda v^2) -\frac{1}{2^\star}\int_\Omega (v+\phi_t^m)_+^{2^\star}\\ &-\int_\Omega G(x, (v+\phi_t^m)_+ ) -\int_\Omega ( f-f_m )v \,, \end{align*} where $H$ denotes the Sobolev space $H=H_0^1(\Omega)$, equipped with the Dirichlet norm $\|u\|=(\int_\Omega |\nabla u|^2\, dx)^{1/2}$. We begin by estimating the error" given by the term $f - f_m$. \begin{lemma}\label{l1} For $N \ge 3$, as $m\to +\infty$ we have: \begin{gather}\label{fm0} \| \phi_t-\phi_t^m \| \le cm^{-N/2};\\ \label{fm} \Big| \int_\Omega ( f-f_m )\psi\,dx \Big| \le c \| \psi \| m^{-N/2}, \quad \mbox{for all } \psi \in H. \end{gather} \end{lemma} \paragraph{Proof.} Note first that by the regularity assumption $h \in L^r, r > N$, it follows that $\phi_t \in C^{1,1-N/r}({\overline \Omega})$, and hence in particular that there exists $c>0$ such that for any point $\bar{x} \in \partial \Omega$ $| \phi_t(x)| \le c|x-\bar{x}| .$ Furthermore we may choose for every large $m \in \mathbb{N}$ a point $x_0$ at distance $4/m$ from the boundary point $\bar{x}$, such that $$\label{ptilde} | \phi_t(x)| \le \frac{ c_1}{m}, \quad \forall x \in B_{4/m}(x_0).$$ We may assume that $x_0=0$ for every choice of $m$; from now on we write $B_r=B_r(0)$. Thus, we can estimate \begin{align*} \int_\Omega | & \nabla ( \phi_t-\phi_t^m ) |^2\\ =& \int_\Omega | \nabla \phi_t (1-\eta_m) -\phi_t \nabla \eta_m|^2\\ =&\int_{B_{\frac{2}{m}}} |\nabla \phi_t|^2|1-\eta_m|^2 -2\int_{B_{\frac{2}{m}}\setminus B_{\frac{1}{m}}} |\nabla \phi_t| ( 1-\eta_m) |\phi_t|\;|\nabla \eta_m| \\ &+\int_{B_{\frac{2}{m}}\setminus B_{\frac{1}{m}}} |\phi_t|^2|\nabla \eta_m|^2 \\ \leq& c_1 m^{-N}+c_2 m^{-N}+c_3 m^{-N} = c m^{-N} , \end{align*} hence (\ref{fm0}). The estimate (\ref{fm}) is now obtained as follows: \begin{align*} \Big| \int_\Omega ( f-f_m )\psi\,dx \Big| =& \Big| \int_\Omega |\nabla (\phi_t - \phi_t^m)\nabla \psi - \lambda (\phi_t - \phi_t^m)\psi dx \Big| \\ \leq& c \|\phi_t -\phi_t^m\| \; \| \psi \| \le c \| \psi \| m^{-N/2} \,, \end{align*} for all $\psi \in H$. \hfill %$\square$ Let $\lambda_1<\lambda_{2}\le \dots$ the eigenvalues of $-\Delta$ and $e_1,\,e_2\dots$, the corresponding eigenfunctions. Take $m$ as before and let $\zeta_m:\;\Omega\rightarrow\mathbb{R}$ be smooth functions such that $0 \le \zeta_m\le 1, |\nabla\zeta_m|\le4m$ and $\zeta_m(x)= \begin{cases} 0& \mbox{if } x\in B_{2/m}\\ 1& \mbox{if } x\in \Omega \setminus B_{3/m} \end{cases}$ We define approximate eigenfunctions" by $e_i^m=\zeta_me_i$. Then the following estimates hold. \begin{lemma}\label{l1a} As $m\to\infty$, we have $e_i^m\to e_i$ in $H$. Moreover, in the space $H^-_{j,m}=\mathop{\rm span}\{e_1^m,\dots,e_j^m\}$, we have $\max \Big\{\|u\|^2: u\in H^-_{j,m},\;\int_\Omega u^2=1\Big\} \le\lambda_j+c_jm^{-N}$ and $\int_\Omega \nabla e_i^m \nabla e_j^m dx = \delta_{ij} + O(m^{-N}).$ \end{lemma} \paragraph{Proof.} See \cite{GR} and observe that, since $\partial \Omega$ is of class $C^1$, also for the eigenfunctions $e_i$ an estimate as (\ref{ptilde}) holds. \hfill %$\square$ \smallskip Consider the family of functions $u_\varepsilon^\star(x)= \Big[ \frac{\sqrt{N(N-2)}\varepsilon}{\varepsilon^2+|x|^2}\Big]^{(N-2)/2}$ which are solutions to the equation \begin{gather*} -\Delta u = |u|^{2^\star-2} u \quad \mbox{in } \mathbb{R}^N \\ u(x) \to 0 \quad \mbox{as } |x| \to \infty \end{gather*} and which realize the best Sobolev embedding constant $H^1( \mathbb{R}^N) \subset L^{2^*}(\mathbb{R}^N)$, i.e. the value $$S=S_N = \inf_{u \not= 0} \frac{\| u \|_{H}}{\|u \|_{L^{2^*}}} \, .$$ Let $\xi\in C^1_0(B_{1/m})$ be a cut--off function such that $\xi(x)=1$ on $B_{{1}/{2m}}$, $0\le\xi(x)\le 1$ in $B_{1/m}$ and $\| \nabla\xi \|_\infty\le 4m$. Let $u_\varepsilon(x):=\xi(x)u_\varepsilon^\star(x) \in H$. For $\varepsilon\to0$ we have the following estimates due to Brezis and Nirenberg \begin{lemma}[Brezis-Nirenberg, \cite{BN}] \label{brn} For fixed $m$ we have \begin{description} \item[(a)] $\|u_\varepsilon\|^2=S^{N/2}+O(\varepsilon^{N-2})$ \item[(b)] $\|u_\varepsilon\|^{2^\star}_{2^\star}=S^{N/2}+O(\varepsilon^{N})$ \item[(c)] $\|u_\varepsilon\|^2_2 \ge K_1\varepsilon^2+O(\varepsilon^{N-2})$ \item[(d)] $\|u_\varepsilon \|_s^s\ge K_2\varepsilon^{N-\frac{N-2}{2}s}$. \end{description} Moreover, for $m \to \infty$ and $\varepsilon=o(1/m)$, we have (see \cite{GR}) \begin{description} \item[(e)] $\|u_\varepsilon\|^2=S^{N/2}+O((\varepsilon m)^{N-2})$ \item[(f)] $\|u_\varepsilon\|^{2^\star}_{2^\star}=S^{N/2}+O((\varepsilon m)^{N})$ \end{description} while (c) and (d) hold independently of $m$. \end{lemma} \paragraph{Proof.} We only prove (d); for the other estimates, see \cite{BN,GR,St}. \begin{align*} \int_{\Omega}u_\varepsilon^s\,dx \ge& c\int_{B_{\frac{1}{2m}}} \big( \frac{\varepsilon}{\varepsilon^2+|x|^2} \big)^{\frac{N-2}{2}s}\,dx \\ \ge & c\int_0^\varepsilon \big( \frac{\varepsilon}{\varepsilon^2+\rho^2} \big)^{\frac{N-2}{2}s} \rho^{N-1}d\rho \ge c\varepsilon ^{-\frac{N-2}{2}s}\varepsilon^N. \end{align*} which completes the proof. \hfill %$\square$ \section{The linking structure} In this section we prove that the functional $J$ has a linking structure" as required by the {\it Generalized Mountain Pass Theorem} by P. Rabinowitz \cite{Ra}. For the rest of this article, we assume $\lambda\in[\lambda_k,\lambda_{k+1})$. Let $H^+ = [\mathop{\rm span} \{ e_1, \dots , e_k \}]^\perp$, $S_r = \partial B_r \cap H^+$, $H^-_{m}=span\{e_1^m,\dots, e_k^m\}$ and $Q_m^\varepsilon = (B_R \cap H_m^-) \oplus [0,R]\{u_\varepsilon\}$, where $m \in \mathbb{N}$ is fixed. Define the family of maps ${\cal H} = \{ h : Q_m^\varepsilon \to H \ \hbox{continuous} : h \big|_{\partial Q_m^\varepsilon } = id \}$, and set $$\label{c} \bar{c} = \inf_{h \in {\cal H}} \sup_{u \in h(Q_m^\varepsilon )} J(u)$$ Then the Generalized Mountain Pass theorem of P. Rabinowitz states that if \begin{quote} 1) $J : H \to \mathbb{R}$ satisfies the Palais-Smale condition (PS) \\ 2) there exist numbers $0 < r < R$ and $\alpha_1 > \alpha_0$ such that \begin{gather}\label{1} J(v) \ge \alpha_1, \quad\mbox{for all } v \in S_r \\ \label{2} J(v) \le \alpha_0, \quad\mbox{for all } v \in \partial Q_m^\varepsilon\,, \end{gather} then the value $\bar{c}$ defined by (\ref{c}) satisfies $\bar{c} \ge \alpha_1$, and it is a critical value for $J$. \end{quote} First note that for $v\in H^-_m\oplus\mathbb{R}\{u_\varepsilon\}$, $v =w+s u_\varepsilon$, we have by definition $\mathop{\rm supp}(u_\varepsilon)\cap\mathop{\rm supp}(w)=\emptyset$. It is easy to prove that this implies that $$J(v)\equiv J(w+s u_\varepsilon)=J(w)+J(s u_\varepsilon) \, .$$ We begin by showing that the functional $J$ satisfies condition (\ref{1}). \begin{lemma}\label{rr0} There exist numbers $r > 0$ and $\alpha_1>0$ such that $J(v)\ge\alpha_1\quad \mbox{for all } v \in S_{r }=\partial B_{r}\cap H^+$ \end{lemma} \paragraph{Proof.} Let $v\in H^+$. From the variational characterization of $\lambda_{k+1}$ and the Sobolev embedding theorem, we have, using ($g_3$) and Lemma \ref{l1}, \begin{eqnarray*} J(v)&\ge & \frac{1}{2} \big(1-\frac{\lambda}{\lambda_{k+1}} \big) \int_\Omega |\nabla v|^2 - \frac{1}{2^\star} \int_\Omega v_+^{2^\star} - c \int_\Omega v_+^{p+1} -\int_\Omega ( f-f_m) v \\ &\ge & c_1\|v\|^2-c_2 \|v \|^{2^\star}-c_3 \|v \|^{p+1}- c_4 m^{-N/2} \|v\| \end{eqnarray*} Let $k_m(s)=c_1 |s|^2- c_2 |s|^{2^\star}-c_3|s|^{p+1}- c_4 m^{-N/2} |s|$. Clearly, there exists $m_0$ such that $\max_{\mathbb{R}}k_m(s)=M_m \ge M_{m_0} > 0$ for all $m \ge m_0$. Thus there exist $\alpha_1 > 0$ and $r > 0$ such that $J(v)\ge \alpha_1 > 0 \quad \mbox{for } \|v\|=r.$ which completes the proof. \hfill %$\square$ Next we prove condition (\ref{2}). \begin{lemma}\label{rr0bis} There exist $R > r$ and $\alpha_0 < \alpha_1$ such that for $\varepsilon$ sufficiently small $J\big|_{\partial Q_m^\varepsilon } < \alpha_0 \,.$ \end{lemma} \paragraph{Proof.} Let $v=w+s u_\varepsilon \in (H_m^- \cap {\bar B}_R) \oplus [0,R]\{u_\varepsilon]$. Since $J(v)=J(w)+J(s u_\varepsilon)$ we can estimate $J(w)$ and $J(s u_\varepsilon)$ separately. $J(w) \le \frac{1}{2} \int_\Omega | \nabla w |^2\,dx -\frac{\lambda}{2} \int_\Omega |w|^2\,dx -\frac{1}{2^\star} \int_\Omega ( w+\phi_t^m)_+^{2^\star}\,dx\,,$ since $\int_\Omega (f - f_m)w\,dx = \int_\Omega \nabla (\phi_t - \phi_t^m)\nabla w \,dx- \lambda \int_\Omega (\phi_t - \phi_t^m)w\,dx =0$. $\begin{array}{lcl} %\begin{multline*} \displaystyle J(s u_\varepsilon) &\le &\displaystyle\frac{s^2}{2} \int _{B_{\frac{1}{m}}}|\nabla u_\varepsilon|^2\,dx -\frac{\lambda s^2}{2}\int _{B_{\frac{1}{m}}} |u_\varepsilon|^2\,dx \\ &-&\displaystyle\frac{s^{2^\star}}{2^\star}\int _{B_{\frac{1}{m}}} |u_\varepsilon |^{2^\star}\,dx - \int _{B_{\frac{1}{m}}}f \; s u_\varepsilon \,dx. \end{array}$ %\end{multline*} Let $\partial Q_m^\varepsilon=\Gamma_1 \cup \Gamma_2 \cup \Gamma_3$, where %\begin{gather*} $\begin{array}{lcl} \Gamma_1 &=& \{v\in H : v=w+su_\varepsilon ,\; w\in H^-_m , \; \|w\|=R,\; 0\le s \le R \},\\ \Gamma_2 &=& \{v\in H : v=w+R u_\varepsilon,\; w\in H^-_m \cap \bar{B}_R \},\\ \Gamma_3 &=& H^-_m \cap \bar{B}_R\,. \end{array}$ %\end{gather*} Note that it follows by Lemma \ref{l1a} that $\int_\Omega | \nabla w |^2 \,dx\le ( \lambda_k + c_k m^{-N}) \int_\Omega |w|^2 \,dx, \quad \hbox{for all } w \in H^-_m$ 1. Suppose $v \in \Gamma_1$; then $v=w+s u_\varepsilon$ with $\| w \| = R \;$ and $\;\, 0 \le s \le R$. \\ (i) if $\lambda \in (\lambda_k,\lambda_{k+1})$, we choose $m_0$ such that $c_k m^{-n}<\frac{\lambda-\lambda_k}{2}$, for $m\ge m_0$. Then, using Lemma \ref{brn} \begin{align*} J(v) \le & \frac{1}{2} \big( 1-\frac{\lambda}{\lambda_k+c_k m^{-N}}\big) \int_\Omega | \nabla w |^2 \,dx -\frac{1}{2^\star} \int_\Omega ( w+\phi_t^m)_+^{2^\star}\,dx\\ & + \frac{s^2}{2}\int_{B_{\frac{1}{m}}}|\nabla u_\varepsilon|^2\,dx -\frac{\lambda s^2}{2}\int_{B_{\frac{1}{m}}} u_\varepsilon^2\,dx -\frac{s^{2^\star}}{2^\star}\int_{B_{\frac{1}{m}}} u_\varepsilon^{2^\star}\,dx +s \| f \|_2 \| u_\varepsilon \|_2 \\ \le& -c R^2 + S^{N/2}\big[\frac{s^2}{2} -\frac{s^{2^\star}}{2^\star}\big] + R^{2^\star} O(\varepsilon^{N})+R^2 O(\varepsilon^{N-2} )+c s \\ \le&-c R^2 +c_1 +c_2 R +c_3 R^{2^\star}\varepsilon^{N-2}\,. \end{align*} Thus $J(v)\le 0$ for $R\ge R_0$ and $\varepsilon >0$ sufficiently small. \\ (ii) if $\lambda=\lambda_k$, for $w=R \bar{w} \in H_m^-$ with $\|\bar{w}\|=1$ we write $\bar{w}=\alpha y+\beta e_k^m$, with $y\in \mathop{\rm span}\{e_1^m,\dots,e_{k-1}^m\}$ and $\|y\|=1$. Then $J(w)=\frac{R^2}{2} \int_\Omega ( |\alpha \nabla y+\beta \nabla e_k^m|^2- \lambda_k|\alpha y+\beta e_k^m|^2) \,dx- \frac{R^{2^\star}}{2^\star} \int_\Omega \big(\bar{w}+\frac{\phi_t}{R} \big)_+^{2^\star} \,dx.$ Using Lemma \ref{l1a}, we can estimate the first integral as follows \begin{eqnarray*} \lefteqn{\int_\Omega ( |\alpha \nabla y+\beta \nabla e_k^m|^2- \lambda_k|\alpha y+\beta e_k^m|^2) \,dx}\\ &\le& \alpha^2\big(1-\frac{\lambda_k}{\lambda_{k-1}+c m^{-N}} \big) +\beta^2\big(1-\frac{\lambda_k}{\lambda_{k}+c m^{-N}} \big) \\ &&+ 2\alpha \beta \int_\Omega ( \nabla y \nabla e_k^m - \lambda_k y e_k^m )\,dx\\ &\le& -c \alpha^2 + c_1 ( \beta^2+2 \alpha \beta ) m^{-N}. \end{eqnarray*} Note now that if $|\alpha|\ge \delta>0$, for some $\delta>0$, then $J(w)\le -\frac{c \delta^2 R^2}{2} + c_1 m^{-N}$ and hence $J(v)\le 0$ for $R\ge R_1(\delta)$. We show now that there exists $\delta > 0$ such that if $|\alpha|\le \delta$, then there exist constants $c_2>0$ and $R_2 > 0$ such that $\int_{\Omega} \big( \bar{w}+\frac{\phi_t}{R}\big)_+^{2^\star}\ge c_2>0$ for all $R \ge R_2$ and for all $w\in H^-_m\cap \partial B_R$. To this aim we prove that there exist $\delta > 0$ and $\eta > 0$ such that $\max_{\bar{\Omega}} ( \alpha y+\beta e_k^m ) \ge \eta > 0 \quad \hbox{for all } y\in H^-_{k-1,m},\,\|y\|=1,\; |\alpha|\le \delta \,.$ By contradiction assume that there exist sequences $|\alpha_n| \le 1/n$, $y_n \in H_{k-1,m}^-$ with $\|y_n\|=1$ such that $\max_{\bar \Omega} \{\alpha_n y_n+\beta_n e_k^m \} \to 0 \quad \mbox{as} \quad n\to+\infty.$ Then $\alpha_n y_n\to 0$, $\beta^2_n=1-\alpha_n^2 + O(m^{-N})\to \beta^2 = 1 + O(m^{-N}) \ge 1/2$, for $m \ge m_0$. Therefore, we conclude that $\max_{\bar \Omega} ( \beta e_k^m) = 0 \ , \mbox{ with } \beta^2 \ge \frac 12 \mbox{ for } m \ge m_0 \mbox{ i.e. }\ ( e_k^m)^+=0 \,.$ This is a contradiction since $e_k^m\to e_k$ in $H$ implies that $e_k^m$ must change sign, for $m$ large. Therefore, there exist $\delta > 0$, $\eta > 0$ such that $\max_{\bar \Omega} \left\{ {\bar w}, |\alpha| \le \delta \right\} \ge \eta > 0 \ , \ \forall {\bar w} \in H_{k,m}^- , \|{\bar w}\| = 1, m \ge m_0 \,.$ Denoting $\Omega_{\bar{w}}=\{ x\in \Omega:\; \bar{w}(x)\ge\ \eta/2 \}$, then $|\Omega_{\bar w}| \ge \nu > 0$, $\forall {\bar w} \in H_{k,m}^-$, with $\|{\bar w}\| = 1$ and $|\alpha|\le \delta$, $m \ge m_0$, since the functions ${\bar w} \in H_{k,m}^-$ are equicontinuous. Moreover $\frac{\phi_t}{R}>-\frac{\eta}{4} \quad \mbox{for }R \mbox{ sufficiently large}\,.$ Then $\int_{\Omega} ( \bar{w}+\frac{\phi_t}{R})_+^{2^\star}\ge ( \frac{\eta}{4} )^{2^\star} | \Omega_{\bar{w}} |.$ Thus, we can conclude that there exists $R_2 > 0$ such that $J(w) \le c R^2 - R^{2^\star} \int_\Omega ({\bar w} + \frac{\phi_t }{R})_+^{2^*} \le c R^2 - R^{2^\star} (\frac{\eta}{4})^{2^*} \; \nu \le 0 \ ,$ for all $R\ge R_2$. In particular $J(w)\to -\infty$ as $R\to +\infty$. \smallskip \noindent 2. Let $v\in\Gamma_2$, i.e. $v = w + R u_\varepsilon$ with $\| w \| \le R$. Then \begin{align*} J(v) =& J(w)+J(R u_\varepsilon)\\ \le& c m^{-N} \| w \|^2 +\frac{R^2}{2}\int_{B_{\frac{1}{m}}}|\nabla u_\varepsilon|^2\,dx -\frac{R^{2^\star}}{2^\star}\int_{\Omega} u_\varepsilon^{2^\star}\,dx + R \| f \|_2 \| u_\varepsilon \|_2 < 0 \,, \end{align*} for $R$ sufficiently large. Now fix $R>0$ such that the previous estimates hold. \smallskip \noindent 3. Let $v \in \Gamma_3$, i.e. $v = w \in H^-_m\cap B_R$. Hence $J(v) \le c_1 m^{-N} \|w\|^2-\frac{1}{2^\star}\int_\Omega( w+\phi_t )_+^{2^\star} \le \alpha_0$ if $m$ is sufficiently large. \hfill %$\square$ \section{Existence of a second solution} In this section we prove Theorems \ref{t1} and \ref{t2}. By the Linking Theorem we construct a Palais--Smale sequence $\{v_n\}\subset H$ at the minimax level $\bar{c}$; the sequence $\{v_n\}$ satisfies \begin{aligned} J(v_n) = &\frac{1}{2}\int_\Omega( |\nabla v_n|^2-\lambda v_n^2 )\,dx -\frac{1}{2^\star}\int_\Omega( v_n+\phi_t^m )_+^{2^\star}\,dx \\ &-\int_\Omega G(x, (v_n+\phi_t^m )_+)\,dx +\int_\Omega (f-f_m)v_n\,dx =\bar{c}+o(1) \end{aligned} \label{p_1} and \begin{aligned} \langle J^\prime(v_n),z\rangle = & \int_\Omega( \nabla v_n \nabla z-\lambda v_n z)\,dx -\int_\Omega( v_n+\phi_t^m )_+^{2^\star-1} z\,dx \\ &-\int_\Omega g(x,(v_n+\phi_t^m )_+)z\,dx +\int_\Omega (f-f_m)z\,dx =o(1)\|z\| \end{aligned} \label{p_2} for all $z\in H$. \newpage \begin{lemma}\label{nn0} Under the hypotheses of Theorem \ref{t1} or Theorem \ref{t2}, the sequence $\{v_n\}$ is bounded in $H$. \end{lemma} \paragraph{Proof.} >From (\ref{p_1}) and (\ref{p_2}), it follows that \begin{eqnarray*} \lefteqn{ J(v_n)-\frac{1}{2}\langle J^\prime(v_n),v_n\rangle }\\ &=& \frac{1}{N}\int_\Omega( v_n+\phi_t^m )_+^{2^\star} -\frac{1}{2} \int_\Omega( v_n+\phi_t^m)_+^{2^\star-1} \phi_t^m\\ && - \int_\Omega G(x, (v_n+\phi_t^m)_+) +\frac{1}{2}\int_\Omega g(x, (v_n+\phi_t^m )_+)v_n - \frac{1}{2}\int_\Omega( f-f_m)v_n\\ &=& \bar{c}+o(1)+o(1)\|v_n\| \end{eqnarray*} Therefore, using that $\phi_t^m\le 0$ and Lemma 4, \begin{eqnarray*} \frac{1}{N} \int_\Omega (v_n+\phi^m_t )_+^{2^\star} \,dx & \le & \int_\Omega G(x, (v_n+\phi_t^m )_+)\,dx \\ &&-\frac{1}{2} \int_\Omega g(x, (v_n+\phi^m_t )_+)(v_n+\phi^m_t)\,dx\\ &&+c +( o(1)+dm^{-N} ) \|v_n\| \end{eqnarray*} Then by ($g_3$), we get \begin{eqnarray*} \lefteqn{\int_\Omega (v_n+\phi^m_t )_+^{2^\star} \,dx}\\ & \le & c_1 \int_\Omega (v_n+\phi^m_t )_+^{p+1} \,dx +{c}+( o(1)+dm^{-N} ) \|v_n\|\\ & \le & c_1 \Big( \int_\Omega (v_n+\phi^m_t )_+^{2^\star} \,dx \Big)^{\frac{p+1}{2^\star}}+c+( o(1)+dm^{-N} ) \|v_n\| \end{eqnarray*} Since $p+1<2^\star$, we obtain $$\int_\Omega (v_n+\phi_t )_+^{2^\star}\,dx \le {c}+(o(1)+dm^{-N} ) \|v_n\| \le c_1+c_2 \|v_n\| \label{pp4}$$ $(i)$ First we consider the case $\lambda\in(\lambda_k,\lambda_{k+1})$. Let $v_n=v_n^+ + v_n^-$ (as in \cite{DJ}), with $v_n^- \in H^-_k=\mathop{\rm span}\{ e_1 \dots e_k\}$ and $v_n^+\in (H^-_k)^{\perp}$. We obtain \begin{eqnarray*} \langle J^{\prime}(v_n),v_n^+\rangle & = & \int_\Omega ( | \nabla v_n^+|^2 -\lambda ( v_n^+)^2 )\,dx -\int_\Omega ( v_n + \phi_t^m )_+^{2^\star -1} v_n^+\,dx \\ & &- \int_\Omega g( x,( v_n + \phi_t^m )_+ ) v_n^+\,dx -\int_\Omega( f-f_m) v_m^+ dx =o(1) \| v_n^+ \| \end{eqnarray*} \newpage \noindent From the variational characterization of $\lambda_{k+1}$ we get, using the H\"{o}lder and Young inequalities and (\ref{pp4}), \begin{eqnarray*} \lefteqn{ \big( 1-\frac{\lambda}{\lambda_{k+1}} \big) \| v_n^+ \|^2}\\ & \le & \int_\Omega ( v_n + \phi_t^m )_+^{2^\star -1} v_n^+ +c\int_\Omega( v_n +\phi_t^m )_+^p | v_n^+| + o(1) \| v_n^+ \| +dm^{-N} \| v_n^+ \|\\ &\le & \varepsilon \Big( \int_\Omega| v_n^+ |^{2^\star} \Big)^{2/2^\star}+ c_\varepsilon( \int_\Omega( v_n +\phi_t^m )_+^{2^\star} )^{\frac{2(2^\star-1)}{2^\star}} \\ && +c\Big(\int_\Omega( v_n +\phi_t^m )_+^{2^\star} \Big)^{p/2^\star} \Big(\int_\Omega | v_n^+ |^{\frac{2^\star}{2^\star-p}} \Big)^{\frac{2^\star-p}{2^\star}}+ c\| v_n^+ \| \\ & \le& \varepsilon \| v_n^+ \|^2 + c_\varepsilon\Big( \int_\Omega( v_n +\phi_t^m )_+^{2^\star} \Big)^{\frac{2(2^\star-1)}{2^\star}}+ \varepsilon\Big(\int_\Omega | v_n^+ |^{\frac{2^\star}{2^\star-p}} \Big)^{2\frac{2^\star-p}{2^\star}} \\ && +c_\varepsilon\big( \int_\Omega\ ( v_n +\phi_t^m )_+^{2^\star} \Big)^{2 p/2^\star}+c\| v_n^+ \| \end{eqnarray*} By (\ref{pp4}) and by the Sobolev embedding theorems, we obtain $$\| v_n^+ \|^2 \le {c} + o(1) \big(\| v_n \|^{\frac{N+2}{N}} +\| v_n \|^{\frac{2 p}{2^\star}} \big) +c \| v_n^+ | | \label{eqmeno}$$ For $v_n^-\in H^-$, we have \begin{eqnarray*} \lefteqn{ ( \frac{\lambda}{\lambda_k}-1 ) \int_\Omega | \nabla v^-_n |^2}\\ &\le& \int_\Omega\ ( v_n +\phi_t^m )_+^{2^\star-1} | v^-_n | + \int_\Omega\ g(x,( v_n +\phi_t^m )_+) | v^-_n | + \int_\Omega |f-f_m| |v_n^-| \end{eqnarray*} In the same way we obtain $$\| v_n^- \|^2_{H^1} \le \bar{c} + o(1) \big( \| v_n \|^{\frac{N+2}{N}} + \| v_n \|^{\frac{2 p}{2^\star}} \big) +c \| v_n^- \| \label{eqpiu}$$ Joining (\ref{eqmeno}) and (\ref{eqpiu}), we find $\|v_n\|^2\le c+c ( \|v_n\|^{\frac{N+2}{N}} + \|v_n\|^{\frac{2p}{2^\star}} )+c \|v_n\|,$ so $v_n$ is bounded in $H$.\smallskip \noindent $(ii)$ If $\lambda=\lambda_k$ we write $v_n=v_n^-+v_n^+ +\beta_n e_k=w_n+\beta_n e_k$, where we denote with $v_n^-$ and $v_n^+$ the projections of $v_n$ onto the subspace $H^-_{k-1}=\mathop{\rm span}\{e_1,\dots,e_{k-1} \}$ and $H^+_k=(H^-_k)^\perp$ respectively. With a similar argument as above we obtain the following estimate $$\|w_n \|^2\le c+c \big( \| v_n \|^{\frac{N+2}{N}}+\| v_n \|^{\frac{2p}{2^\star}}\big) +c\| w_n \| \label{pp5}$$ We can assume $\|v_n\|\ge 1$. Then, from (\ref{pp5}), we have $$\| w_n \|^2\le c+c \| v_n \|^{\frac{2p}{2^\star}}\le c+c ( \| w_n \|+| \beta_n |)^{\frac{2p}{2^\star}} \label{pstar}$$ If $\beta_n$ is bounded we conclude as above. If not, we may assume $\beta_n \to+\infty$ and $\|w_n\|\to+\infty$ (we neglect the case $\|w_n\|\le c$ which is much easier). Therefore, from (\ref{pstar}), $\| w_n \|^2\le \big[ \frac{1}{2} ( \| w_n \|+\beta_n^2)\big]^{p/2^\star}.$ Then $\| w_n \|\le c \beta_n^{p/2^\star}$ and $\| \frac{w_n}{\beta_n}\|\to 0$ since $\frac{p}{2^\star}<1$. Therefore, possibly up to a subsequence, $w_n/\beta_n\to 0$ a.e. and strongly in $L^q$, $2\le q<2^\star$. Therefore, for all $q\in (2,2^\star)$ $$\int_\Omega ( \frac{w_n+\phi_t^m}{\beta_n}+e_k )^q_+ e_k\,dx \quad\to\quad \int_\Omega( e_k )^{q+1}_+ \,dx. \label{pint}$$ Moreover, since \begin{eqnarray*} o(1)& =&\langle J^\prime(v_n), e_k\rangle \\ &=& -\int_\Omega (v_n+\phi_t^m)^{2^\star-1}_+ e_k -\int_\Omega g(x,(v_n+\phi_t^m )_+) e_k +\int_\Omega (f-f_m) e_k \end{eqnarray*} we get, using $(g_3)$ $\int_\Omega \big( \frac{w_n+\phi_t^m}{\beta_n}+e_k\big) ^{2^\star-1}_+ e_k\le o(1)+\frac{c}{\beta_n^{2^\star-1-p}} \int_\Omega \big( \frac{w_n+\phi_t^m}{\beta_n}+e_k\big)_+^p e_k.$ Finally, by (\ref{pint}) we get $\int_\Omega (e_k )_+^{2^\star}\le 0$ which is a contradiction. Thus $(v_n)$ is bounded. \hfill %$\square$ Returning to relation (\ref{p_2}), we may therefore assume, as $n\to +\infty$: $v_n \rightharpoonup v$ weakly in $H^1_0$, $v_n \to v$ in $L^q \quad 2\le q< 2^\star$ and $v_n \to v$ a. e. in $\Omega$. In particular, it follows that $v$ is a weak solution of $$\begin{gathered} -\Delta v =\lambda v+( v +\phi_t^m )_+^{2^\star-1}+g(x,( v +\phi_t^m )_+) +f-f_m \quad\mbox{in } \Omega\\ v =0 \quad \mbox{on } \partial \Omega \end{gathered}\label{eqcz}$$ To conclude the proof, It remains to show that $v \ne \phi_t-\phi_t^m$, the trivial'' solution of (\ref{eqcz}). First, we estimate \begin{eqnarray*} \lefteqn{ J(\phi_t-\phi_t^m) }\\ &= &\frac{1}{2}\int_\Omega | \nabla (\phi_t-\phi_t^m)|^2 -\frac{\lambda}{2}\int_\Omega| \phi_t-\phi_t^m|^2 -\int_\Omega(f-f_m)(\phi_t-\phi_t^m) \\ &= & -\frac{1}{2} \int_\Omega \big[ | \nabla (\phi_t-\phi_t^m)|^2 -\lambda |\phi_t-\phi_t^m|^2 \big]. \end{eqnarray*} \vspace{5mm} From Lemma \ref{l1} we get, taking $\varepsilon^\beta=\frac{1}{m}\quad 0<\beta<1$, $$|J(\phi_t-\phi_t^m)|\le c m^{-N}:=c \varepsilon^{\beta N}. \label{eqplus}$$ Since $v$ is a weak solution of (\ref{eqcz}), we have \begin{eqnarray*} \int_\Omega ( | \nabla v |^2 -\lambda v^2)\,dx - \int_\Omega ( v +\phi_t^m )_+^{2^\star}\,dx + \int_\Omega ( v +\phi_t^m )_+^{2^\star-1} \phi_t^m\,dx&&\\ -\int_\Omega g(x,( v +\phi_t^m )_+) v\,dx -\int_\Omega ( f-f_m ) v\,dx&=&0 \end{eqnarray*} By the Brezis--Lieb Lemma (\cite{BL}) $$\int_\Omega ( v_n +\phi_t^m )_+^{2^\star}\,dx = \int_\Omega ( v_n -v )_+^{2^\star}\,dx + \int_\Omega ( v +\phi_t^m )_+^{2^\star}\,dx +o(1) \label{eq1st}$$ Since $v_n \to v$ in $L^q \quad 2\le q < 2^\star$, we have $$\int_\Omega G(x,( v_n +\phi_t^m )_+)\,dx - \int_\Omega G(x,( v +\phi_t^m )_+)\,dx =o(1). \label{eq2st}$$ Since $v_n \rightharpoonup v$ in $H$, $$\int | \nabla v_n | ^2 = \int | \nabla v |^2+\int| \nabla (v_n-v )|^2+o(1). \label{eq3st}$$ Then, by (\ref{p_1}), (\ref{eq1st}) and (\ref{eq3st}), we find \begin{eqnarray} \bar{c}+o(1)& =&J( v_n ) \label{p2palle} \\ & =& J( v )+\frac{1}{2} \int_\Omega |\nabla( v_n -v )|^{2}\,dx - \frac{1}{2^\star} \int_\Omega ( v_n -v )_+^{2^\star}\,dx +o(1) \nonumber \end{eqnarray} Similarly, since $J^\prime(v)=0$, we obtain \begin{eqnarray*} \langle J^\prime( v_n ),v_n\rangle & =& \int_\Omega |\nabla( v_n -v)|^{2} - \int_\Omega ( v_n -v )_+^{2^\star} - \int_\Omega ( v_n -v)_+^{2^\star-1} \phi_t^m \\ &&- \int_\Omega g(x,( v_n +\phi_t^m)_+) v_n + \int_\Omega g(x,( v +\phi_t^m)_+) v +o(1). \end{eqnarray*} Since $\int_\Omega ( v_n -v )_+^{2^\star-1} \phi_t^m\,dx=o(1)$ and $\int_\Omega g(x,( v_n +\phi_t^m )_+) v_n\,dx - \int_\Omega g(x,( v +\phi_t^m )_+) v\,dx =o(1),$ we get $$\int_\Omega |\nabla( v_n -v )|^{2}\,dx= \int_\Omega ( v_n -v )_+^{2^\star}\,dx +o(1). \label{ptilde2}$$ Now, let $K=\lim_{n\to+\infty} \int_\Omega |\nabla( v_n -v )|^{2}\,dx.$ If $K=0$, then $v_n\to v$ strongly in $H$ and in $L^{2^\star}$; then by (\ref{p2palle}) and (\ref{eqplus}) $J(v)=\bar{c}\ge \alpha_1>c \varepsilon^{\beta N} \ge J ( \phi_t - \phi_t^m )$ so that $v\ne \phi_t-\phi_t^m$. If $K>0$, using the Sobolev inequality and (\ref{ptilde2}), we have (as in \cite{DJ}) \begin{eqnarray*} \| v_n -v \|^{2} & \ge & S \Big( \int_\Omega | v_n-v|^{2^\star}\,dx \Big)^{2/2^\star} \ge S \Big( \int_\Omega ( v_n-v)_+^{2^\star}\,dx \Big)^{2/2^\star} \\ & \ge & S \Big( \int_\Omega |\nabla ( v_n-v)|^{2}\,dx+o(1) \Big)^{2/2^\star} \end{eqnarray*} This implies that $K\ge S K^{\frac{N-2}{N}}$, that is $K\ge S^{N/2}$. To complete the proof we use the following Lemmas which will be proved below. \begin{lemma}\label{ma1} Under the hypotheses of Theorem \ref{t1} one has, for $\varepsilon^\beta=1/m$, with $\alpha/N < \beta< (N-4)/(N-2)$, $$\bar{c} < \frac {1}{N} S^{N/2} -c \varepsilon^2. \label{pp2}$$ \end{lemma} \begin{lemma}\label{ma2} Suppose that the hypotheses of Theorem \ref{t2}2 are satisfied. Then for $\varepsilon^\beta=1/m$, with $\max \big\{ 1-\frac{N-2}{2N}(q+1), \frac{2(N-1)}{N-2}-(q+1) \big\} <\beta<\frac{1}{2}(q+1)-\frac{2}{N-2},$ we have $\bar{c} < \frac{1}{N} S^{N/2}-c \varepsilon^{N-\frac{N-2}{2}(q+1)}.$ \end{lemma} By (\ref{p2palle}) and (\ref{ptilde2}) we get $J(v)+\frac{K}{N} =\bar{c} \le \frac{1}{N}S^{N/2}- \begin{cases} c \varepsilon^2 & \mbox{(Theorem \ref{t1})}\\ c \varepsilon^{N-\frac{N-2}{2}(q+1)} &\mbox{(Theorem \ref{t2})} \end{cases}$ Assume now by contradiction that $v\equiv \phi_t-\phi_t^m$. Then we get by (\ref{eqplus}) $\frac{K}{N} + J(v) \ge \frac{1}{N}S^{N/2}-c \varepsilon^{\beta N}$ which is impossible, due to the choice of $\beta$. \hfill %$\square$ \paragraph{Proof of Lemma \ref{ma1}.} Let $\varepsilon>0$ and $m \in \mathbb{N}$ be fixed such that the hypotheses of the Linking Theorem are satisfied. For $v=w+s u_\varepsilon$, we have \begin{eqnarray*} J(v) &= &J(w) + J(s u_\varepsilon) \\ &\le& J(w) + \frac{s^2}{2}\int_{B_{\frac{1}{m}}}| \nabla u_\varepsilon |^2 - \frac{s^{2^\star}}{2^\star}\int_{B_{\frac{1}{m}}}u_\varepsilon^{2^\star} -\frac{\lambda s^2}{2}\int_{B_{\frac{1}{m}}}u_\varepsilon^{2}\\ &&+ s \int_{B_{\frac{1}{m}}} |f - f_m|\; |u_\varepsilon | \end{eqnarray*} As above, we have $J(w)\le c m^{-N}$. To estimate the last inequality we use the argument developed in \cite{GR} (Lemma \ref{l1}). We have from ({\it{e}}) and ({\it{f}}) in Lemma \ref{brn} \begin{eqnarray*} \frac{s^2}{2} \int_{B_{1/m}} | \nabla u_\varepsilon |^2 - \frac{s^{2^\star}}{2^\star} \int_{B_{1/m}} u_\varepsilon ^{2^\star} &\le&( \frac{s^2}{2}- \frac{s^{2^\star}}{2^\star} ) \big[ S^{N/2}+O( (\varepsilon m)^{N-2} ) \big] \\ &\le& ( \frac{1}{2}- \frac{1}{2^\star} ) \big[ S^{N/2}+O( (\varepsilon m)^{N-2} ) \big] \\ &\le&\frac{S^{N/2}}{N} + c (\varepsilon m)^{N-2}. \end{eqnarray*} (since $\frac{s^2}{2}- \frac{s^{2^\star}}{2^\star}$ attains its maximum at $s=1$) We make use of the H\"{o}lder inequality to estimate the last term. Let $\alpha$ be such that $\frac{1}{\alpha}+\frac{1}{r}+\frac{1}{2}=1$, i.e. $\frac{1}{\alpha} = \frac{1}{2}-\frac{1}{r} > \frac{1}{2}-\frac{1}{N}=\frac{1}{2^\star}$, in particular $2^\star>\alpha$. Since $\mathop{\rm supp} f_m \subseteq \Omega \backslash B_{1/m}$ we have \begin{eqnarray*} \int_{B_{1/m}} | f-f_m |\, | u_\varepsilon | &\le& \Big( \int_{B_{1/m}} | f |^r \Big)^{1/r} \Big( \int_{B_{1/m}} | u_\varepsilon |^2 \Big)^{1/2} ( \mu ( B_{1/m} ) )^{{1}/{\alpha}}\\ &\le& c \varepsilon m^{-N/\alpha}. \end{eqnarray*} If $\varepsilon^\beta=1/m$ with $0<\beta<1$ we have, by ({\it {c}}) in Lemma \ref{brn}, \begin{aligned} J(v) \le & \frac{1}{N} S^{N/2} + c_1 m^{-N}+ c_2 (\varepsilon m)^{N-2}-c_3 \varepsilon^2+ c_4 \varepsilon m^{N/\alpha} \\ = & \frac{1}{N} S^{N/2} + c_1 \varepsilon^{N \beta} +c_2 \varepsilon^{(1- \beta)(N-2)} -c_3 \varepsilon^2 + \varepsilon^{1+ \beta N/\alpha} \end{aligned} \label{ri3} We choose $\beta$ such that $\frac{\alpha}{N} < \beta < 1 - \frac{2}{N-2}\,.$ Such a choice is possible only for $N \ge 6$. This then implies that for $\varepsilon > 0$ sufficiently small $J(v)<\frac{1}{N} S^{N/2} - c \varepsilon^2,$ in particular $\bar{c}<\frac{1}{N} S^{N/2} - c \varepsilon^2$. \hfill %$\square$ \paragraph{Proof of Lemma \ref{ma2}.} With a similar argument as above and using ($g_4$) and Lemma \ref{brn}(\textit{d}), we obtain \begin{eqnarray*} J(v)& = & J(w) +J(s u_\varepsilon)\\ &\le& J(w) + \frac{s^2}{2}\int_\Omega | \nabla u_\varepsilon|^2\,dx -\frac{\lambda s^2}{2}\int_\Omega u_\varepsilon^2\,dx -\frac{s^{2^\star}}{2^\star}\int_\Omega u_\varepsilon^{2^\star} \,dx\\ &&-\int_\Omega G(x,( s u_\varepsilon +\phi_t^m)_+ )\,dx -s \int_\Omega ( f-f_m) u_\varepsilon\,dx \end{eqnarray*} Reasoning as in Lemma \ref{ma1} from ({\it{d}}) in Lemma \ref{brn} we obtain as in (\ref{ri3}) \begin{eqnarray*} J(v)& \le & \frac{S^{N/2}}{N}+c_1 \varepsilon^{N\beta} +c_2 \varepsilon^{(1-\beta)(N-2)} - c_3 s^{q+1} \int_\Omega u_\varepsilon^{q+1}\,dx + c_4 \varepsilon^{1+\beta N/\alpha}\\ & \le& \frac{S^{N/2}}{N}+c_1 \varepsilon^{N\beta} +c_2 \varepsilon^{(1-\beta)(N-2)} -c_3 \varepsilon^{-\frac{N-2}{2}(q+1)+N} + c_4 \varepsilon^{1+\beta N/\alpha} \end{eqnarray*} Using that $\alpha<2^\star$, we get the following condition on $\beta\in(0,1)$ $N-\frac{N-2}{2} (q+1)< \begin{cases} \beta N \\ 1+\frac{\beta N}{2^\star}\\ (1-\beta) (N-2) \end{cases}$ which is equivalent to the system \begin{gather*} 1-\frac{N-2}{2N}(q+1) < \beta < - \frac{2}{N-2} + \frac{1}{2}(q+1)\\ \frac{2(N-1)}{N-2}- (q+1) < \beta < - \frac{2}{N-2} + \frac{1}{2}(q+1) \end{gather*} This choice is possible if $q+1 > \begin{cases} \frac{N^2}{(N-1)(N-2)}\\ \frac{2}{3} \cdot \frac{2N}{N-2} \end{cases}$ Therefore the result follows. \hfill %$\square$ \section{Existence of a third solution} We consider only the case $N\ge 6$ and $g\equiv 0$; the other cases follow with small changes. Let $\phi_t(\lambda)$ denote the negative solution of (1) for $\lambda\in (\lambda_k-\delta,\lambda_k)$. Since $h\in \ker (-\Delta-\lambda_k)^\perp$, $\phi_t(\lambda)$ is uniformly bounded when $\lambda\to \lambda_k$. Consider the functional $J_\lambda(v)= \frac{1}{2} \int_\Omega ( |\nabla v |^2-\lambda v^2 )\,dx -\frac{1}{2^\star} \int_\Omega ( v+\phi_t(\lambda))_+^{2^\star}\,dx -\int_\Omega ( f-f_m )v\,dx.$ We prove the main geometrical properties of the functional $J_\lambda$. \begin{lemma}\label{le7} Let $H^+=\mathop{\rm span}\{e_1,\dots,e_k\}^\perp$. There exist $\bar{\alpha}>0$, $\bar{r}>0$ such that $J_\lambda(v)\ge\bar{\alpha}>0\quad\mbox{for all } v\in S_{\bar{r}}=\partial B_{\bar{r}}\cap H^+, \; \lambda\in (\lambda_{k-1},\lambda_k ).$ \end{lemma} \paragraph{Proof.} With a similar argument as in Lemma \ref{rr0}, since $\lambda<\lambda_k$, we have \begin{eqnarray*} J_\lambda(v)&\ge& \frac{1}{2} \big( 1-\frac{\lambda}{\lambda_{k+1}} \big) \|v\|^2 -c_2 \|v\|^{2^\star}-c_3 m^{-N/2} \|v\| \\ & \ge& \frac{1}{2} \big( 1-\frac{\lambda_k}{\lambda_{k+1}} \big) \|v\|^2 -c_2 \|v\|^{2^\star}-c_3 m^{-N/2} \|v\| \end{eqnarray*} Then there exist $m_0$, $\bar{\alpha}$ and $\bar{r}$ such that for $m\ge m_0$, $J_\lambda(v)\ge \bar{\alpha}>0$ for $\|v\|=\bar{r}$ and all $\lambda\in( \lambda_{k-1},\lambda_k )$. \hfill %$\square$ Now let $H_m^-=\mathop{\rm span} \{e_1^m,\dots,e^m_k\}$ as in the proof of Theorems \ref{t1} and \ref{t2}. \begin{lemma}\label{ma3} Let $Q_m=(H^-_m\cap B_R(0))\oplus[0,R] u_\varepsilon$, where $\varepsilon^\beta=1/m$. Then, if $R$ is large enough, there exists $\bar{m}\ge m_0$ such that:\\ (i) $J_{\partial Q_{\bar{m}}}<\bar{\alpha}$\\ (ii) $\sup_{Q_{\bar{m}}} J<\frac{S^{N/2}}{N}-c \bar{\varepsilon}^2$. \end{lemma} \paragraph{Proof.} (i) Let $\partial Q_{\bar{m}}=\Gamma_1\cup\Gamma_2\cup\Gamma_3$ as in Lemma \ref{rr0bis}. \noindent (1) For $v=w+s u_\varepsilon\in\Gamma_1$ we have $J(v)=J(w)+J(s u_\varepsilon)$. Arguing as in Lemma \ref{rr0bis} (ii) we have $$J(s u_\varepsilon) \le c_3 R^{2^\star} \varepsilon^{N-2}+c_1+c R \label{eqr1}$$ Writing $w=R \bar{w}\in H_m^-$ with $\| \bar{w} \|=1$, $\bar{w}=\alpha y+\beta e^m_k$, $y\in \mathop{\rm span}\{e_1^m,\dots,e_{k-1}^m\}$, $\|y\|=1$, it was shown in Lemma \ref{rr0bis} (ii) that there exists a number $\sigma>0$ such that for $|\alpha|^2\le \sigma$, $$J(w) \le c_1 R^2-c_2 R^{2^\star}. \label{eqr2}$$ On the other hand, if $|\alpha|^2 > \sigma$, one deduces as in Lemma \ref{rr0bis} (ii) (using that $\delta < \lambda_k-\lambda_{k-1}$) that \begin{eqnarray*} J(w) & \le & \frac{R^2}{2} \int ( \alpha\nabla y+\beta\nabla e^m_k )^2 -\lambda | \alpha y+\beta e^m_k|^2 \\ & \le & \frac{R^2}{2} \Big[ \alpha^2 ( 1-\frac{\lambda}{\lambda_{k-1}+c m^{-N}} ) +\beta^2 ( 1-\frac{\lambda}{\lambda_{k}+c m^{-N}} ) \\ & & +2 \alpha \beta \int ( \nabla y \nabla e^m_k - \lambda y e^m_k ) \Big] \\ & \le & \frac{R^2}{2} \Big[ \sigma \frac{\lambda_{k-1}-( \lambda_k-\delta ) +c m^{-N}}{\lambda_{k-1}+c m^{-N}} +\beta^2 \frac{\delta+c m^{-N}}{\lambda_k+c m^{-N}} +2 \alpha \beta c m^{-N} \Big] \\ & \le & R^2 [-c \sigma+c_1 \delta+c_2 m^{-N}]. \end{eqnarray*} Thus, we get in this case, for $\delta$ sufficiently small and $m$ sufficiently large $$J(w)\le -\frac{c}{2} \sigma R^2. \label{eqr3}$$ Joining (\ref{eqr1}), (\ref{eqr2}) and (\ref{eqr3}) we have $J(v)=J(w)+J(s u_\varepsilon) \le 0,$ for $\delta$ small, $R$ sufficiently large and $\varepsilon$ sufficiently small. \noindent (2) Let $v\in \Gamma_2$, i.e. $v=w+R u_\varepsilon$ with $\|w\|\le R$; then \begin{eqnarray*} J(v) & = & J(w)+J(R u_\varepsilon) \le c ( m^{-N}+\delta) R^2 +\frac{R^2}{2} \int_{B_{\frac{1}{m}}} | \nabla u_\varepsilon|^2 \,dx \\ & & - \frac{R^{2^\star}}{2^\star} \int_{B_{\frac{1}{m}}} u_\varepsilon^{2^\star} \, dx+ R \| f \| \, \| u_\varepsilon \| <0 \end{eqnarray*} for $R$ sufficiently large. \noindent (3) Let $v\in \Gamma_3=\bar{B}_R\cap H^-_m$. Then \begin{eqnarray*} J(v) & \le& c_1 ( m^{-N}+\delta) \|w\|^2 -\frac{1}{2^\star} \int_{B_{\frac{1}{m}}} ( w+\phi_t(\lambda) )^{2^\star}_+ \,dx \\ & \le& c_1 ( m^{-N}+\delta) \|w\|^2 < \alpha_0 \end{eqnarray*} if $m\ge m_1$ and $\delta>0$ sufficiently small. \noindent ii) For $v \in Q_m$ we have as in Lemma \ref{ma1} inequality (\ref{ri3}), $J(v) \le \frac{1}{N} S^{N/2} +c_1 ( m^{-N}+\delta) R^2 +c_2 (\varepsilon m )^{N-2}-c_3 \varepsilon^2-c_4 \varepsilon m^{-N/2}$ If $\varepsilon^\beta=1/m$, with $\beta$ as in Lemma \ref{ma1}, then $J(v)\le \frac{S^{N/2}}{N}-c {\varepsilon}^2+c_4 \delta R^2$ Then there exists $\delta_1$ such that if $0<\delta<\delta_1$, then $J(v)\le \frac{S^{N/2}}{N}-\frac{c}{2} {\varepsilon}^2.$ Arguing as in Theorem \ref{t1}, we find a critical point $v_\lambda$ of the functional $J_\lambda$ at a level $c_\lambda\ge\bar{\alpha}>0$ with $\bar{\alpha}$ independent of $\lambda\in(\lambda_k-\delta,\lambda_k)$. It was shown in \cite[Theorem 1.3]{DJ} that in every $\lambda_k, k\ge 1$, starts a bifurcation branch $(\lambda,k)$ emanating from the negative solution and bending to the left (Prop. 4.2) (and thus corresponding" to our second solution). We conclude that for $\lambda$ to the left and close to $\lambda_k$ there exist at least three solutions for equation (\ref{p1}), for $t>T_0$ with $T_0=T_0(h)$ sufficiently large. \hfill %$\square$ \begin{thebibliography}{00} \frenchspacing \bibitem{AP} A. Ambrosetti, G. Prodi, {\it {On the inversion of some differentiable mappings with singularities between Banach spaces}}, Ann. Mat. Pura Appl., {\bf 93}, 1972, 231--247. \bibitem{BL} H. Brezis, E. Lieb, {\it A relation between pointwise convergence of functions and convergence of integrals}, Proc. Amer. Math. Soc., {\bf 88}, 1983, 486--490. \bibitem{BN} H. Brezis, L. Nirenberg, {\it Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents}, Comm. Pure Appl. Math., {\bf 24}, 1983, 437--477. \bibitem{D} D.G. De Figueiredo, {\it On superlinear elliptic problems with nonlinearities interacting only with higher eigenvalues}, Rocky Mountain J. Math., {\bf 18}, 1988, 287--303. \bibitem{D1} D.G. De Figueiredo, {\it Lectures on boundary values problems of Ambrosetti--Prodi type}, Atas do $12^\circ$ Seminario Brasileiro de An\'{a}lise, S\~{a}o Paulo, 1980. \bibitem{DJ} D.G. De Figueiredo, Y. Jianfu, \textit{Critical superlinear Ambrosetti--Prodi problems}, Topological Methods in Nonlinear Analysis, {\bf 14}, 1999, 59--80. \bibitem{DR} D.G. De Figueiredo, B. Ruf, {\it On a superlinear Sturm--Liouville equation and a related bouncing problem}, J. reine angew. Math. {\bf 421} (1991), 1--22. \bibitem{GR} F. Gazzola, B. Ruf, {\it Lower order perturbations of critical growth nonlinearities in semilinear elliptic equations}, Advances in Diff. Eqs., {\bf 2}, n. 4, 1997, 555-572. \bibitem{GT} D. Gilbarg, N.S. Trudinger, {\it Elliptic partial differential equations of second order}, Springer--Verlag, New York, 1993. \bibitem{Ra} P. Rabinowitz, {\it Minimax methods in critical point theory with applications to differential equations}, 65 AMS Conf. Sec. Math, 1986. \bibitem{RS} B. Ruf, B., P.N. Srikanth, {\it Multiplicity results for superlinear elliptic problems with partial interference with spectrum}, J. Math. Anal. App., {\bf 118}, 1986, 15--23. \bibitem{St} M. Struwe, {\it Variational methods}, Springer--Verlag, Berlin, 1996. \end{thebibliography} \noindent\textsc{Marta Calanchi} (e-mail: calanchi@mat.unimi.it, ph. +39.02.50316144)\\ \textsc{Bernhard Ruf} (e-mail: ruf@mat.unimi.it, ph. +39.02.50316157)\\[3pt] Dip. di Matematica, Universit\`{a} degli Studi,\\ Via Saldini 50, 20133 Milano, Italy \end{document}