\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 124, pp. 1--17.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/124\hfil Bifurcation for the $p$-Laplacian] {Bifurcation along curves for the $p$-Laplacian with radial symmetry} \author[F. Genoud \hfil EJDE-2012/124\hfilneg] {Fran\c cois Genoud} \address{Fran\c cois Genoud \newline Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom} \email{F.Genoud@hw.ac.uk} \thanks{Submitted March 7, 2012. Published July 31, 2012.} \thanks{This work was supported by the Engineering and Physical Sciences Research Council, \hfill\break\indent [EP/H030514/1].} \subjclass[2000]{35J66, 35J92, 35B32} \keywords{Dirichlet problem; radial $p$-Laplacian; bifurcation; solution curve} \begin{abstract} We study the global structure of the set of radial solutions of a nonlinear Dirichlet eigenvalue problem involving the $p$-Laplacian with $p>2$, in the unit ball of $\mathbb{R}^N$, $N \geqslant 1$. We show that all non-trivial radial solutions lie on smooth curves of respectively positive and negative solutions, bifurcating from the first eigenvalue of a weighted $p$-linear problem. Our approach involves a local bifurcation result of Crandall-Rabinowitz type, and global continuation arguments relying on monotonicity properties of the equation. An important part of the analysis is dedicated to the delicate issue of differentiability of the inverse $p$-Laplacian, and holds for all $p>1$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this article we consider the nonlinear Dirichlet problem $$\label{dir.eq} \begin{gathered} -\Delta_p (u) = \lambda f(|x|,u) \quad \text{in } \Omega,\\ u =0 \quad \text{on } \partial \Omega, \end{gathered}$$ where $\Delta_p(u):=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian, $p>1$, $\lambda \geqslant0$, and $\Omega$ is the unit ball in $\mathbb{R}^N$, $N\geqslant1$. The function $f$ is continuous, such that $f(r,0)=0$ for all $r\in[0,1]$, and will be subject to various additional assumptions. We look for $C^1$ radial solutions by studying the problem \label{dirad.eq} \begin{gathered} -(r^{N-1}\phi_p(u'))' = \lambda r^{N-1} f(r,u), \quad 00 \quad\text{and}\quad f_\infty(r):=\lim_{|\xi|\to\infty}\frac{f(r,\xi)}{\phi_p(\xi)}>0, \quad r\in[0,1]. $$The properties of \eqref{eigen} with f_0 enable us to obtain a local bifurcation theorem as in \cite{gs}, while the asymptotic problem \eqref{eigen} with f_\infty governs the behaviour as |u|\to\infty. We will consider two different situations. In the first case, we will assume that f(r,\xi)>0 for all (r,\xi)\in[0,1]\times\mathbb{R}^* and f(r,0)=0 for all r\in[0,1]. It follows that the set of non-trivial solutions of \eqref{dirad.eq} is a smooth curve of positive solutions --- see Theorem~\ref{main.thm}. If we rather assume f(r,\xi)\xi>0 for (r,\xi)\in [0,1]\times\mathbb{R}^* and f(r,0)\equiv0, then we get two smooth curves of respectively positive and negative solutions, containing all non-trivial solutions of \eqref{dirad.eq} --- see Theorem~\ref{main2.thm}. Furthermore, if N=1, we are also able to deal with the case where f is sublinear' at infinity, that is, f(r,\xi)/\phi_p(\xi)\to0 as |\xi|\to\infty, uniformly for r\in[0,1]. A one-dimensional problem similar to \eqref{dirad.eq} was studied by Rynne in \cite{r10}, from which the present work is substantially inspired. In particular, the explicit form we get for the inverse p-Laplacian in the radial setting allows us to study the differentiability of this operator following arguments of \cite{br}, where the one-dimensional p-Laplacian was considered. This differentiability issue is probably the most delicate part of the analysis. It should be noted that the results regarding the inverse p-Laplacian in Section~\ref{inverse.sec} hold for any p>1, while we had to restrict ourselves to p>2 in the bifurcation analysis for other reasons --- see Remark~\ref{diff.rem}. We conclude this section with a brief description of the contents of the paper. In Section~\ref{results.sec}, we give some information about the functional setting, our precise hypotheses, and we state our main results, Theorems~\ref{main.thm} and \ref{main2.thm}. Then, in Section~\ref{inverse.sec}, we study an integral operator corresponding to the inverse of the p-Laplacian in \eqref{dirad.eq}. The main results about this operator are Theorems~\ref{Spcont.thm} and \ref{Sp.thm}. It should be noted that \cite{gs} already dealt with differentiability results similar to those of Theorem~\ref{Sp.thm}. However, we believe that the discussion in \cite{gs} is incomplete and so Theorem~\ref{Sp.thm} is of importance in its own right. In Section~\ref{prop.sec}, we establish some {\it a priori} properties of solutions of \eqref{dirad.eq}, notably positivity/negativity, as well as the asymptotic behaviour of solutions (\lambda,u) as |u|\to0/\infty. Section~\ref{cranrab.sec} is devoted to the local bifurcation analysis, where we establish, in particular, a Crandall-Rabinowitz-type result, Lemma~\ref{cranrab.lem}. Finally, the proofs of Theorems~\ref{main.thm} and \ref{main2.thm} are completed in Section~\ref{global.sec}, where we show that the local branches of solutions obtained in Section~\ref{cranrab.sec} can be extended globally. \section{Setting and main results}\label{results.sec} We will work in various function spaces. We will denote by L^1(0,1) the Banach space of real Lebesgue integrable functions over (0,1) and by W^{1,1}(0,1) the Sobolev space of functions u\in L^1(0,1) having a weak derivative u' \in L^1(0,1). For n=0,1, C^n[0,1] will denote the space of n times continuously differentiable functions, with the usual sup-type norm |\cdot|_n. In our operator formulation of \eqref{dirad.eq}, it will be convenient to use the shorthand notation$$ X_p := \{u\in C^1[0,1] : \phi_p(u') \in C^1[0,1] \ \text{and} \ u'(0)=u(1)=0\} , \quad Y:=C^0[0,1]. $$An important part of our discussion in the next section will concern the differentiability of an integral operator, that will depend on the value of p>1. This analysis will rely on results in \cite{br}, and we borrow the following notation from there: $$\label{bp.eq} B_p := \begin{cases} C^1[0,1], & 1< p \leqslant 2,\\ W^{1,1}(0,1), & p>2. \end{cases}$$ However, our main results require p\geqslant2 and, apart from Section~\ref{inverse.sec}, we will suppose p>2 throughout the paper --- the results are well known for p=2. Denoting by \partial_2f the partial derivative of f with respect to \xi\in\mathbb{R}, we make the following hypotheses on the continuous function f:[0,\infty)\times\mathbb{R}\to\mathbb{R}: \begin{itemize} \item[(H1)] f(r,\cdot)\in C^1(\mathbb{R}) for all r\in [0,1] and \partial_2f \in C^0([0,1]\times\mathbb{R}); \item[(H2)] f(r,\xi)>0 for (r,\xi) \in [0,1]\times\mathbb{R}^* and f(r,0)\equiv0; \item[(H3)] (p-1)f(r,\xi) \geqslant \partial_2f(r,\xi)\xi for (r,\xi) \in [0,1]\times[0,\infty), and there exist \delta,\epsilon>0 such that (p-1)f(r,\xi) > \partial_2f(r,\xi)\xi for all (r,\xi)\in(1-\delta,1]\times (0,\epsilon). \end{itemize} It follows from (H3) that, for any fixed r\in[0,1], the mapping \xi \mapsto f(r,\xi)/\phi_p(\xi) is decreasing on (0,\infty). Therefore, for each r\in[0,1] there exist f_0(r) and f_\infty(r) such that$$f(r,\xi)/\phi_p(\xi) \to f_{0/\infty}(r) \quad\text{as} \ \xi \to 0^+/+\infty,$$with $$\label{gbounds} 0\leqslant f_\infty(r) \leqslant f(r,\xi)/\phi_p(\xi) \leqslant f_0(r), \quad\text{for all} \ (r,\xi) \in [0,1]\times(0,\infty).$$ We will further suppose that f_0(r) is finite for all r\in[0,1], that f_0, f_\infty \in C^0[0,1], and (for p>2): \begin{itemize} \item[(H4)] \lim_{\xi\to0^+}|f(\cdot,\xi)/\phi_p(\xi) - f_0|_0 = \lim_{\xi\to0^+}|\partial_2 f(\cdot,\xi)/\xi^{p-2} - (p-1) f_0|_0 = 0; \item[(H5)] \lim_{\xi\to\infty}|f(\cdot,\xi)/\phi_p(\xi) - f_\infty|_0 = 0. \end{itemize} \begin{remark}\label{sign.rem}\rm \quad \begin{itemize} \item[(i)] Note that (H2) and \eqref{gbounds} imply f_0>0 on [0,1]. \item[(ii)] Also, (H3) implies that, for r\in(1-\delta,1], the function \xi \mapsto f(r,\xi)/\phi_p(\xi) is not constant on (0,\infty). In particular, f_\infty \not\equiv f_0 on [0,1]. \end{itemize} \end{remark} To state our main results, we need to relate problem \eqref{dirad.eq} to the homogeneous eigenvalue problems corresponding to the asymptotes f_0, f_\infty \in C^0[0,1]: \label{eigen} %\tag{\mathrm{E}_{0/\infty}} \begin{gathered} -(r^{N-1}\phi_p(v'))' = \lambda r^{N-1} f_{0/\infty}(r)\phi_p(v), \quad 00 on [0,1] then problem \eqref{eigen} has a simple eigenvalue \lambda_{0/\infty}>0 with a corresponding eigenfunction v_{0/\infty}>0 in [0,1), and no other eigenvalue having a positive eigenfunction. Furthermore, f_\infty \substack{\leqslant \\ \not\equiv} f_0 implies \lambda_0<\lambda_\infty. \end{lemma} We know from Remark \ref{sign.rem} that f_0>0 and 0\leqslant f_\infty \ \substack{\leqslant \\ \not\equiv} \ f_0. For \lambda_\infty to be well-defined, we will still make the following assumption. \begin{itemize} \item[(H6)] Either\\ (a) N\geqslant1 is arbitrary and f_\infty>0 on [0,1], or\\ (b) N=1 and f_\infty\equiv0 on [0,1].\\ If (a) holds, \lambda_\infty>0 is defined in Lemma~\ref{eigen.lem}; if (b) holds, we set \lambda_\infty=\infty. \end{itemize} We are now in a position to state our first result about the solutions of \eqref{dirad.eq}. From now on, we will refer to the collection of hypotheses (H1) to (H6) as (H). \begin{theorem}\label{main.thm} Suppose that p > 2. If {\rm (H)} holds, there exists u\in C^1((\lambda_0,\lambda_\infty), Y) such that u(\lambda) \in X_p, u(\lambda)>0 on [0,1) and, for any \lambda \in (\lambda_0,\lambda_\infty), (\lambda,u(\lambda)) is the unique non-trivial solution of \eqref{dirad.eq}. Furthermore, $$\label{asympt.eq} \lim_{\lambda\to\lambda_0} |u(\lambda)|_0=0 \quad\text{and}\quad \lim_{\lambda\to\lambda_\infty} |u(\lambda)|_0=\infty.$$ \end{theorem} We will see in the proof of Theorem~\ref{main.thm} that the condition (H2) forces the solutions of \eqref{dirad.eq} to be positive. If, instead of (H2) to (H5), we suppose: \begin{itemize} \item[(H2')] f(r,\xi)\xi>0 for (r,\xi) \in [0,1]\times\mathbb{R}^* and f(r,0)\equiv0; \item[(H3')] in addition to (H3), (p-1)f(r,\xi) \leqslant \partial_2f(r,\xi)\xi for (r,\xi) \in [0,1]\times(-\infty,0] and (p-1)f(r,\xi) < \partial_2f(r,\xi)\xi for all (r,\xi)\in(1-\delta,1]\times (-\epsilon,0); \item[(H4')] \lim_{\xi\to0}|f(\cdot,\xi)/\phi_p(\xi) - f_0|_0 = \lim_{\xi\to0}|\partial_2 f(\cdot,\xi)/|\xi|^{p-2} - (p-1) f_0|_0 = 0; \item[(H5')] \lim_{|\xi|\to\infty}|f(\cdot,\xi)/\phi_p(\xi) - f_\infty|_0 = 0, \end{itemize} then the solutions need not be positive any more and we have the following result. We refer to the collection of hypotheses (H1), (H2') to (H5') and (H6) as (H'). \begin{theorem}\label{main2.thm} Suppose that p > 2. If {\rm (H')} holds, there exist u_\pm\in C^1((\lambda_0,\lambda_\infty), Y) such that u_\pm(\lambda) \in X_p, \pm u_\pm(\lambda)>0 on [0,1) and, for any \lambda \in (\lambda_0,\lambda_\infty), (\lambda,u_\pm(\lambda)) are the only non-trivial solutions of \eqref{dirad.eq}. Furthermore, both u_- and u_+ satisfy the limits in \eqref{asympt.eq}. \end{theorem} We will prove Theorems~\ref{main.thm} and \ref{main2.thm} by giving the detailed arguments for the case where (H) holds, and explaining what needs to be modified to account for (H'). \begin{remark}\rm Theorems~\ref{main.thm}-\ref{main2.thm} yield a complete description of the set of solutions of \eqref{dirad.eq}, hence the set of radial solutions of \eqref{dir.eq}. Under the additional assumption that f(r,\xi) is non-increasing in r\in[0,1], it follows from \cite[Theorem~1,~p.~51]{bro} that positive solutions u\in W_0^{1,p}(\Omega)\cap C^1(\overline{\Omega}) of \eqref{dir.eq} are radial, and hence satisfy \eqref{dirad.eq}. Thus in this case, Theorems~\ref{main.thm}-\ref{main2.thm} characterize all positive solutions of \eqref{dir.eq}. \end{remark} As immediate corollaries of Theorems~\ref{main.thm} and \ref{main2.thm}, we have the following results of existence and uniqueness/multiplicity for the problem \label{diradf.eq} \begin{gathered} -(r^{N-1}\phi_p(u'))' = r^{N-1} f(r,u), \quad 02, suppose that {\rm (H)} holds, and that 1\in(\lambda_0,\lambda_\infty). Then problem \eqref{diradf.eq} has a unique non-trivial solution u\in X_p, such that u>0 on [0,1). \end{corollary} \begin{corollary}\label{cor2} Let p>2, suppose that {\rm (H')} holds, and that 1\in(\lambda_0,\lambda_\infty). Then problem \eqref{diradf.eq} has exactly two non-trivial solutions u_\pm\in X_p, satisfying \pm u_\pm>0 on [0,1). \end{corollary} \subsection*{Notation} For h\in C^0([0,1]\times\mathbb{R}), we define the Nemitskii mapping h:Y\to Y by h(u)(r):=h(r,u(r)). Then h:Y\to Y is bounded and continuous. We will always use the same symbol for a function and for the induced Nemitskii mapping. When computing estimates, the symbol C will denote positive constants which may change from line to line. Their exact values are not essential to the analysis. \section{The inverse p-Laplacian}\label{inverse.sec} In this section we study an integral operator corresponding to the inverse of the p-Laplacian in the radial setting. Although our main goal in this paper is the proof of Theorems~\ref{main.thm} and \ref{main2.thm}, which requires p\geqslant2, the results in this section will be stated in greater generality, for p>1. Let us start by introducing some useful notation. For p>1, we let$$ p^*:=\tfrac{1}{p-1} \quad\text{and}\quad p':=\tfrac{p}{p-1}=p^*+1. Then for \xi,\eta\in\mathbb{R}, the continuous function \phi_p:\mathbb{R}\to\mathbb{R} satisfies $$\label{phi.eq} \phi_p(\xi)=\eta \iff \xi=\phi_{p'}(\eta)=\phi_{p^*+1}(\eta).$$ For any h\in C^0[0,1], the problem \label{heq.eq} \begin{gathered} -(r^{N-1}\phi_p(u'))' = r^{N-1} h(r), \quad 01; \\ %\label{Phi.eq}\notag \mathcal{I}:C^0[0,1] \to C^1[0,1], \quad \mathcal{I}(k)(r) &:= \int_r^1 k(s) \,\mathrm{d} s;\\ %\label{defofI.eq}\notag T_p:C^0[0,1] \to C^1[0,1], \quad T_p &:=\mathcal{I}\circ\Phi_{p'} \quad\text{for any } p>1. \end{align*} It is clear that \Phi_q, \mathcal{I} and T_p are continuous and bounded, for any q,p>1, and that \mathcal{I} is linear. Furthermore, S_p is p^*-homogeneous. \begin{remark} \rm For 12 (i.e. p'<2). Nevertheless, if g\in C^1[0,1] has only simple zeros, then \Phi_q maps a neighbourhood of g in C^1[0,1] continuously into L^1(0,1) (see Lemma~2.1 in \cite{br}). \end{remark} The following lemma gives important properties of J. \begin{lemma}\label{J.lem} \quad \begin{itemize} \item[(i)] J: C^0[0,1] \to C^1[0,1] is well-defined. \item[(ii)] J: C^0[0,1] \to C^1[0,1] is a bounded linear operator, with norm \Vert J\Vert \leqslant 2. \item[(iii)] J is Fr\'echet differentiable on C^0[0,1] with DJ=J. \item[(iv)] J: C^0[0,1] \to C^0[0,1] is compact. \end{itemize} \end{lemma} \begin{proof} (i) Using de l'Hospital's rule, we get \lim_{s\to0} J(s)=\tfrac{1}{N-1}\lim_{s\to0}sh(s)=0, $$and it follows that J(h)\in C^0[0,1] for all h\in C^0[0,1]. Furthermore,$$ \lim_{s\to0}\frac{J(s)}{s}=\lim_{s\to0}\frac{s^{N-1}h(s)}{Ns^{N-1}}=\frac{h(0)}{N}, $$and so J(h)\in C^1[0,1], with J'(0)=h(0)/N. (ii) We first have \begin{equation*}\label{estJ0.eq} |J(h)|_0 \leqslant \sup_{0\leqslant s\leqslant 1} \Big| \int_0^s \Big(\frac{t}{s}\Big)^{N-1} \,\mathrm{d} t \Big| |h|_0 \leqslant \frac{|h|_0}{N}. \end{equation*} Then, since $$\label{diffJ.eq} \begin{split} J(h)'(s) &=(1-N)s^{-N}\int_0^s t^{N-1} h(t) \,\mathrm{d} t + h(s) \\ &=(1-N)s^{-1}J(h)(s)+h(s) \quad \text{for all} \ s\in(0,1], \end{split}$$ we have \begin{equation*}\label{estJ1.eq} |J(h)'|_0 \leqslant \sup_{0\leqslant s\leqslant 1} \big| (1-N)s^{-N} \int_0^s t^{N-1} \,\mathrm{d} t \big| |h|_0 + |h|_0 \leqslant \frac{2N-1}{N} |h|_0. \end{equation*} It follows that J is bounded, with norm$$ \Vert J\Vert:=\sup_{|h|_0=1} |J(h)|_1\leqslant \frac1N + \frac{2N-1}{N}=2. $$(iii) follows from (ii). (iv) follows from (ii) and the compact embedding C^1[0,1] \hookrightarrow C^0[0,1]. \end{proof} We can now state important properties of S_p, following from the above results. \begin{theorem}\label{Spcont.thm} The mapping S_p: C^0[0,1] \to C^1[0,1] defined by \eqref{comp.eq} is continuous, bounded and compact. \end{theorem} The following result is a simple adaptation of \cite[Theorem~3.2]{br} to the present context. This is the first step towards the differentiability of S_p. \begin{proposition}\label{Tp.prop} \quad \begin{itemize} \item[(i)] Suppose 12 and let g_0\in C^1[0,1] have only simple zeros (i.e. g_0(s)=0 \Rightarrow g_0'(s)\neq0). Then T_p: C^1[0,1] \to B_p is C^1 on a neighbourhood U_0 of g_0 in C^1[0,1], and \eqref{derivofTp.eq} holds for all g\in U_0, \ \bar g \in C^1[0,1]. \end{itemize} \end{proposition} We can now prove the main result of this section, about the differentiability of S_p. Its statement and proof are very similar to those of Theorem~3.4 in \cite{br}. \begin{theorem}\label{Sp.thm} \quad \begin{itemize} \item[(i)] Suppose 12 and let h_0\in C^0[0,1] be such that u(h_0)'(r)=0 \Rightarrow h_0(r)\neq0. Then there exists a neighbourhood V_0 of h_0 in C^0[0,1] such that the mapping h\mapsto |u(h)'|^{2-p}: V_0 \to L^1(0,1) is continuous, S_p:V_0 \to B_p is C^1, and DS_p satisfies \eqref{DSp.eq} and \eqref{lineariz.eq}, for all h\in V_0, \bar h \in C^0[0,1]. \end{itemize} \end{theorem} \begin{proof} (i) In view of \eqref{comp.eq}, the differentiability of S_p: C^0[0,1] \to B_p follows from Lemma~\ref{J.lem}(iii) and Proposition~\ref{Tp.prop}(i). Then, for h,\bar h \in C^0[0,1],$$ DS_p(h)\bar h = DT_p(J(h))J(\bar h) = p^*\mathcal{I}(|J(h)|^{p^*-1}J(\bar h)). $$Now letting u(h)=S_p(h) and differentiating \eqref{formula.eq} yields$$ u(h)'=\phi_{p'}(J(h))=\phi_{p^*+1}(J(h)) \Longrightarrow |u(h)'|^{2-p}=|J(h)|^{p^*-1}, $$proving \eqref{DSp.eq}, from which the continuity of DS_p follows. We will prove below that \eqref{lineariz.eq} holds in both cases (i) and (ii). (ii) The case p>2 is more delicate and uses Proposition~\ref{Tp.prop}(ii). We define g_0:=J(h_0)\in C^1[0,1] and u_0:=u(h_0). Then$$ -\phi_p(u_0'(r))=g_0(r) \quad\text{and}\quad -(r^{N-1}\phi_p(u_0'(r)))' =r^{N-1}h_0(r), \quad 0\leqslant r\leqslant 1. $$We will show that g_0 has only simple zeros. First remark that$$ g_0(r)=0 \Longrightarrow \phi_p(u_0'(r))=0 \Longrightarrow u_0'(r)=0 \Longrightarrow h_0(r)\neq 0, $$by our hypothesis. But now by \eqref{diffJ.eq}, $$\label{dg0.eq} g_0'(r)=(1-N)r^{-1}g_0(r)+h_0(r), \quad 0\leqslant r\leqslant 1.$$ Therefore, if g_0(r)=0 with r>0, then g_0'(r)=h_0(r)\neq0. On the other hand, if g_0(0)=0, it follows from \eqref{dg0.eq} that Ng_0'(0)=h_0(0)\neq0. Hence, g_0 has only simple zeros. Apart from our statement \eqref{lineariz.eq}, which is slightly more precise than its analogue in \cite{br}, the proof then follows that of \cite[Theorem~3.4]{br}, using Proposition~\ref{Tp.prop}(ii) and the analogue of \cite[Lemma~2.1]{br} for the present setting. To prove statement \eqref{lineariz.eq}, let v=DS_p(h)\bar h, h,\bar h \in C^0[0,1]. Then, from \eqref{DSp.eq},$$ v(r)=p^*\int_r^1|u(h)'(s)|^{2-p}\int_0^s\Big(\frac{t}{s}\Big)^{N-1}\bar h(t) \,\mathrm{d} t \,\mathrm{d} s, \quad r\in[0,1]. $$Since |u(h)'|^{2-p} \in L^1(0,1) in both cases (i) and (ii), it follows that v(1)=0. Furthermore, $$\label{flat} v'(r)=-p^*|u(h)'(r)|^{2-p}\int_0^r\Big(\frac{t}{r}\Big)^{N-1}\bar h(t) \,\mathrm{d} t, \quad r\in[0,1] ,$$ from which the equation in \eqref{lineariz.eq} easily follows. But \eqref{flat} also implies$$ |v'(r)|\leqslant C_1 |u(h)'(r)|^{2-p}r, \quad r\in[0,1]. $$Now since u(h)'(r)=-\phi_{p'}(\int_0^r(t/r)^{N-1} h(t) \,\mathrm{d} t), it follows that |u(h)'(r)|\leqslant C_2 r^{p'-1}, so that$$ |v'(r)|\leqslant C r^{(p'-1)(2-p)+1} = C r^{p^*}, \quad r \in [0,1], $$showing that v'(0)=0 and finishing the proof. \end{proof} \begin{remark}\rm Note that Theorem~\ref{Sp.thm} reduces to well-known results for p=2. \end{remark} \section{Properties of solutions}\label{prop.sec} In this section we discuss some {\it a priori} properties of solutions. We first study the sign of solutions and then we determine their behaviour as |u|_0\to0/\infty. By the results of Section~\ref{inverse.sec}, (\lambda,u) \in [0,\infty)\times X_p is a solution of \eqref{dirad.eq} if and only if $$\label{dirop.eq} F(\lambda,u) := u - \lambda^{p^*} S_p(f(u)) = 0, \quad (\lambda,u)\in [0,\infty)\times Y.$$ Note that F:[0,\infty)\times Y\to Y is continuous. Furthermore, F(0,u)=0 \implies u=0, so we will only consider solutions in$$ \mathcal{S} := \{(\lambda,u) \in (0,\infty) \times Y: (\lambda,u) \text{ is a solution of \eqref{dirop.eq} with }u\not\equiv0 \}. $$\subsection{The case where (H) holds} We start with the positivity of solutions. \begin{proposition}\label{pos.prop} Let (\lambda,u) \in \mathcal{S}. Then u>0 on [0,1), u is decreasing and satisfies u'(1)<0. \end{proposition} \begin{proof} Equation \eqref{dirop.eq} yields$$ u(r)=\lambda^{p^*}\int_r^1\phi_{p'}\Big(\int_0^s \Big(\frac{t}{s}\Big)^{N-1} f(t,u(t)) \,\mathrm{d} t\Big) \,\mathrm{d} s. $$Since u\not\equiv0 is continuous, it follows from (H2) that u(0)>0. Furthermore,$$ \phi_p(u'(r)) = -\lambda \int_0^r\Big(\frac{t}{r}\Big)^{N-1} f(t,u(t)) \,\mathrm{d} t \leqslant 0, \quad r \in [0,1], $$showing that u'(r) \leqslant 0 for all r\in[0,1], so u is decreasing on [0,1]. Finally,$$ \phi_p(u'(1)) = -\lambda \int_0^1 t^{N-1} f(t,u(t)) \,\mathrm{d} t < 0. $$This implies u'(1)<0, from which u>0 on [0,1) now follows. \end{proof} For the following results, we will use the function g:[0,1]\times\mathbb{R}\to\mathbb{R} defined by $$\label{gdef} g(r,\xi) := \begin{cases} f(r,\xi)/\phi_p(\xi) , \quad \xi\neq0,\\ f_0(r), \quad \xi=0. \end{cases}$$ It follows from our assumptions that g \in C^0([0,1]\times\mathbb{R}) and g satisfies $$\label{boundsforg} 0\leqslant f_\infty(r) \leqslant g(r,\xi) \leqslant f_0(r), \quad (r,\xi) \in [0,1]\times\mathbb{R}.$$ \begin{lemma}\label{asympt.lem} Consider a sequence \{(\lambda_n,u_n)\} \subset \mathcal{S}. Suppose that |u_n|_0\to0/\infty as n\to\infty. Then \lambda_n\to\lambda_{0/\infty}. \end{lemma} \begin{proof} Setting v_n:=u_n/|u_n|_0, we have $$\label{eqforvn} v_n=S_p(\lambda_ng(u_n)\phi_p(v_n))$$ or, equivalently, \label{slforvn} \begin{gathered} -(r^{N-1}\phi_p(v_n'))' = \lambda_n r^{N-1} g(u_n)\phi_p(v_n), \quad 00 in [0,1) for all n, it follows from \eqref{boundsforg}, \eqref{slforvn}, and the Sturmian-type comparison theorem in \cite[Sec. 4]{w} that $$\label{lambounds.eq} 0<\lambda_0 \leqslant \lambda_n \leqslant \lambda_\infty\leqslant\infty.$$ Let us first suppose that hypothesis (H6)(a) holds. Then \lambda_\infty<\infty and we can suppose that \lambda_n\to\bar\lambda \in [\lambda_0,\lambda_\infty] as n\to\infty. Now |v_n|_0=1 for all n, so \{\lambda_ng(u_n)\phi_p(v_n)\} is bounded in C^0[0,1] and \{S_p(\lambda_ng(u_n)\phi_p(v_n))\} is bounded in C^1[0,1]. Therefore, by \eqref{eqforvn}, we can suppose that |v_n-\bar v|_0 \to 0 as n\to\infty, for some \bar v \in C^0[0,1]. It then follows by fairly standard arguments (see e.g. the proof of \cite[Lemma~5.4]{d}) that$$ g(u_n)\phi_p(v_n) \to f_{0/\infty}\phi_p(\bar v) \quad \text{provided } |u_n|_0\to0/\infty. $$Hence \bar v satisfies \bar v=S_p(\bar\lambda f_{0/\infty}\phi_p(\bar v)) if |u_n|_0\to0/\infty; that is, \label{slforvbar} \begin{gathered} -(r^{N-1}\phi_p(\bar v'))' = \bar\lambda r^{N-1} f_{0/\infty}\phi_p(\bar v), \quad 00 in [0,1) and it follows from the properties of the eigenvalue problem \eqref{slforvbar} (see \cite[Sec.~5]{w}) that \bar\lambda=\lambda_{0/\infty}. We next suppose that hypothesis (H6)(b) holds; i.e., N=1 and f_\infty\equiv0. We first prove that \lambda_n\to\infty if |u_n|_0\to\infty. Indeed, if we suppose instead that \{\lambda_n\} is bounded, then the above argument yields a \bar v \in C^0[0,1] such that v_n \to \bar v in C^0[0,1] (up to a subsequence), and it follows that g(u_n)\phi_p(v_n)\to0 in C^0[0,1]. Then \eqref{eqforvn} implies \bar v=0, contradicting |\bar v|_0=1. Regarding the behaviour as |u_n|_0\to0, the argument for the case f_\infty>0 will hold in exactly the same way for f_\infty\equiv0 if we can show that \{\lambda_n\} is bounded. It follows from \eqref{eqforvn} that$$ 1 =|v_n|_0=v_n(0)= \lambda_n^{p^*} \int_0^1 \phi_{p'} \Big( \int_0^s g(u_n)\phi_p(v_n)\,\mathrm{d} t\Big) \,\mathrm{d} s. Since u_n, v_n are decreasing on [0,1] and, for any fixed t\in[0,1], the mapping \xi \mapsto f(t,\xi)/\xi^{p-1} is decreasing on (0,\infty), we have \begin{align*} \lambda_n^{-p^*} & =\int_0^1 \phi_{p'} \Big( \int_0^s g(u_n)\phi_p(v_n)\,\mathrm{d} t\Big) \,\mathrm{d} s \\ & \geqslant \int_0^{1/2} \phi_{p'} \Big( \int_0^s \phi_p(v_n(1/2))\frac{f(t,u_n(0))}{u_n(0)^{p-1}} \,\mathrm{d} t\Big) \,\mathrm{d} s \\ & \geqslant Cv_n(1/2)\min_{0\leqslant t \leqslant\frac12} \Big(\frac{f(t,u_n(0))}{u_n(0)^{p-1}}\Big)^{p^*}. \end{align*} Since N=1, it follows from \eqref{dirad.eq} that u is concave for all (\lambda,u)\in\mathcal{S}. Hence, there is a constant M>0 (independent of n) such that v_n(1/2)\geqslant M|v_n|_0=M for all n. Furthermore, f(t,u_n(0))/u_n(0)^{p-1}\to f_0(t)>0 uniformly for t\in[0,\frac12] and so there exists \delta>0 such that \lambda_n^{-p^*}\geqslant\delta for n large enough. Therefore, \{\lambda_n\} is bounded. Note that the arguments above only show that \lambda_{n_k}\to\lambda_{0/\infty} for a subsequence \{\lambda_{n_k}\}. Since they can be applied to any subsequence of \{\lambda_n\}, it follows that the whole sequence must converge. This concludes the proof. \end{proof} \begin{remark}\label{lambounds.rem}\rm The proof of \eqref{lambounds.eq} shows that \lambda_0\leqslant\lambda\leqslant\lambda_\infty for all (\lambda,u)\in\mathcal{S}. \end{remark} \subsection{The case where (H') holds} In this case we consider solutions in the sets \mathcal{S}^\pm := \{(\lambda,u) \in (0,\infty) \times Y: (\lambda,u) \ \text{is a solution of} \ \eqref{dirop.eq} \ \text{with} \ u\not\equiv0, \ \text{and} \ \pm u \geqslant0 \}. $$The following result can be proved as Proposition~\ref{pos.prop}, using (H2') instead of (H2). \begin{proposition}\label{pos'.prop} Let (\lambda,u) \in \mathcal{S}^\pm. Then \pm u>0 on [0,1), \pm u is decreasing and satisfies \pm u'(1)<0. \end{proposition} Regarding the asymptotic behaviour, we have \begin{lemma}\label{asympt'.lem} Consider a sequence \{(\lambda_n,u_n)\} \subset \mathcal{S}^\pm. Suppose that |u_n|_0\to0/\infty as n\to\infty. Then \lambda_n\to\lambda_{0/\infty}. \end{lemma} \begin{proof} The proof is the same as for Lemma~\ref{asympt.lem} in the case where \{(\lambda_n,u_n)\} \subset \mathcal{S}^+. In case u_n\leqslant0, as similar proof can be carried out, setting v_n:=-u_n/|u_n|_0\geqslant0 and remarking that, with this new definition, v_n still satisfies \eqref{eqforvn}. \end{proof} \begin{remark}\label{lambounds'.rem}\rm We also have \lambda_0\leqslant\lambda\leqslant\lambda_\infty for all (\lambda,u)\in\mathcal{S}^\pm. \end{remark} \section{Local bifurcation}\label{cranrab.sec} To prove Theorems~\ref{main.thm} and \ref{main2.thm}, we begin with a local bifurcation result in the spirit of Crandall and Rabinowitz \cite{cr}. This will allow us to start off the bifurcating branch from the line of trivial solutions at the point (\lambda_0,0) in \mathbb{R}\times Y. Crandall and Rabinowitz' original result pertained to semilinear equations, i.e. p=2. A first generalization to p>2 was given in \cite{gs} for a problem very similar to \eqref{dirad.eq}. The main difference in our setting is that we allow the asymptote f_0 to depend on r, so that we get the weighted eigenvalue problem \eqref{eigen} with f_0 instead of problem \eqref{hom}. In the following, we assume the principal eigenfunction v_0 given in Lemma~\ref{eigen.lem} is normalized so that$$ \int_0^1 r^{N-1} f_0(r) |v_0(r)|^p \,\mathrm{d} r = 1. $$We define the subspace$$ Z:=\{z \in Y: \int_0^1 r^{N-1} f_0 |v_0|^{p-2}v_0 z \,\mathrm{d} r = 0\} $$and we remark that $$\label{decomp} Y=\mathrm{span}\{v_0\}\oplus W.$$ To be able to discuss later cases (H) and (H'), it will be convenient to state our local bifurcation result more generally, in terms of the function G: \mathbb{R}^2 \times Z \to Y defined by$$ G(s,\lambda,z):= \begin{cases} v_0+z - S_p(\lambda f(sv_0+sz)/\phi_p(s)), & s\neq0,\\ v_0+z - S_p(\lambda f_0\phi_p(v_0+z)), & s=0. \end{cases} $$Note that G(s,\lambda,z)=F(\lambda,s(v_0+z))/s for all s\neq0, where F:\mathbb{R}\times Y \to Y was defined in \eqref{dirop.eq}. It follows from the definitions of \lambda_0 and v_0 that G(0,\lambda_0,0)=0. \begin{lemma}\label{cranrab.lem} Let p>2 and suppose that (H1) and (H4) hold. There exist \varepsilon>0, a neighbourhood U of (\lambda_0,0) in \mathbb{R}\times Z and a continuous mapping s \mapsto (\lambda(s),z(s)): (-\varepsilon,\varepsilon) \to U such that (\lambda(0),z(0))=(\lambda_0,0) and$$ \{(s,\lambda,z)\in(-\varepsilon,\varepsilon) \times U: G(s,\lambda,z)=0\} =\{(s,\lambda(s),z(s)):s\in(-\varepsilon,\varepsilon)\}. $$\end{lemma} \begin{proof} Our proof follows that of \cite[Theorem~1]{gs} but we give it here for completeness. Under hypotheses (H1) and (H4), it is easily seen that G is continuous. It follows from Theorem~\ref{Sp.thm}(ii) that S_p is C^1 in a neighbourhood of \lambda_0 f_0 \phi_p(v_0) in Y. A routine verification then shows that there is a neighbourhood U of (0,\lambda_0,0) in \mathbb{R}^2\times Z such that the mapping (\lambda,z) \mapsto G(s,\lambda,z) is differentiable in$$ A_s:=\{(\lambda,z) \in \mathbb{R}\times Z: (s,\lambda,z) \in U\}, $$for any s\in\mathbb{R} such that A_s \neq\emptyset. Furthermore, the Fr\'echet derivative D_{(\lambda,z)} G is continuous on U. Since $$\label{opeigen} v_0=S_p(\lambda_0 f_0 \phi_p(v_0)),$$ it follows from \eqref{DSp.eq} that $$\label{derivG.eq} \begin{split} & D_{(\lambda,z)}G(0,\lambda_0,0)(\bar\lambda,\bar z) \\ &= \bar z - \lambda_0(p^*)^{-1} DS_p(\lambda_0 f_0 \phi_p(v_0))f_0|v_0|^{p-2} \bar z -p^* (\bar\lambda/\lambda_0)v_0, \quad (\bar\lambda, \bar z) \in \mathbb{R} \times Z. \end{split}$$ To conclude the proof using the implicit function theorem as stated in Appendix~A of \cite{cr}, we need only check that D_{(\lambda,z)}G(0,\lambda_0,0): \mathbb{R} \times Z \to Y is an isomorphism. Let us first show that the mapping$$ L\bar z := \lambda_0(p^*)^{-1} DS_p(\lambda_0 f_0 \phi_p(v_0))f_0|v_0|^{p-2} \bar z $$leaves the subspace Z invariant. Suppose \bar z \in Z and let z=L\bar z. By \eqref{lineariz.eq} and \eqref{opeigen}, we have \label{eqforz} \begin{gathered} -(r^{N-1}|v_0'|^{p-2}z')' =\lambda_0 r^{N-1} f_0|v_0|^{p-2}\bar z, \quad 01, we first had some hope to obtain bifurcation for all p>1. It turns out that the integration by parts arguments involved in the proof of Lemma~\ref{cranrab.lem} require at least p\geqslant 1+1/N (for the boundary terms to vanish). Moreover, the differentiability of the function G in the present functional setting requires p\geqslant2, and we have not been able to find another suitable setting allowing for p<2. \end{remark} We can now state the local bifurcation results for equation \eqref{dirop.eq}. \begin{theorem}\label{loc1.thm} Let p>2 and suppose that {\rm (H)} holds. There exist \varepsilon_0\in(0,\varepsilon) and a neighbourhood U_0 of (\lambda_0,0) in \mathbb{R}\times Y such that $$\label{loc1.eq} \{(\lambda,u) \in U_0: F(\lambda,u)=0\}=\{(\lambda(s),s(v_0+z(s))): s \in [0,\varepsilon_0)\}.$$ \end{theorem} \begin{proof} It follows from Lemma~\ref{cranrab.lem} that (\lambda(s),s(v_0+z(s))) is a solution of \eqref{dirop.eq} for all s \in [0,\varepsilon). To prove the reverse inclusion in \eqref{loc1.eq}, let us first remark that, by Proposition~\ref{pos.prop}, s\in(-\varepsilon,0) yields no solutions of \eqref{dirop.eq}. Furthermore, a compactness argument similar to that in \cite[p.~39]{gs} shows that any solution in a small enough neighbourhood of (\lambda_0,0) in \mathbb{R}\times Y must have the form (\lambda(s),s(v_0+z(s))) for some s \in [0,\varepsilon). This completes the proof. \end{proof} \begin{theorem}\label{loc2.thm} Let p>2 and suppose that {\rm (H')} holds. There exist \varepsilon_1\in(0,\varepsilon) and a neighbourhood U_1 of (\lambda_0,0) in \mathbb{R}\times Y such that $$\label{loc2.eq} \{(\lambda,u) \in U_1: F(\lambda,u)=0\}=\{(\lambda(s),s(v_0+z(s))): s \in (-\varepsilon_1,\varepsilon_1)\},$$ with $$\label{loc2-.eq} \{(\lambda(s),s(v_0+z(s))): s \in (-\varepsilon_1,0)\}\subset\mathcal{S}^-$$ and $$\label{loc2+.eq} \{(\lambda(s),s(v_0+z(s))): s \in (0,\varepsilon_1)\}\subset\mathcal{S}^+.$$ \end{theorem} \begin{proof} The local characterization of solutions in \eqref{loc2.eq} follows similarly to \eqref{loc1.eq} in Theorem~\ref{loc1.thm}. For \varepsilon_1>0 small enough, statements \eqref{loc2-.eq} and \eqref{loc2+.eq} follow from the construction of the solutions (\lambda(s),s(v_0+z(s))). \end{proof} \section{Global continuation}\label{global.sec} Our goal in this final section is to complete the proofs of Theorems~\ref{main.thm} and \ref{main2.thm}. Namely, we will first show that the local curves of solutions obtained in Section~\ref{cranrab.sec} can be parametrized by \lambda and then we will prove that they can be extended globally. \subsection{Proof of Theorem~\ref{main.thm}} We begin with a non-degeneracy result implying that, in fact, through any non-trivial solution of \eqref{dirop.eq}, there passes a (local) continuous curve of solutions, parametrized by \lambda. \begin{lemma}\label{IFT.lem} The function F \in C([0,\infty)\times Y, Y) defined in \eqref{dirop.eq} is continuously differentiable in a neighbourhood of any point (\lambda,u) \in \mathcal{S}, with$$ D_u F(\lambda,u) v = v - \lambda DS_p(\lambda f(u)) \partial_2 f(u) v, \quad v\in Y. $$Furthermore, for any (\lambda,u) \in \mathcal{S}, D_u F(\lambda,u):Y\to Y is an isomorphism. \end{lemma} \begin{proof} The statement about the differentiability of F follows from Theorem~\ref{Sp.thm}(ii) and Proposition~\ref{pos.prop}. Furthermore, we see that D_u F(\lambda,u):Y\to Y is a compact perturbation of the identity. Therefore, to show that it is an isomorphism, we only need to prove that N(D_u F(\lambda,u))=\{0\}. Let v\in N(D_u F(\lambda,u)). By \eqref{lineariz.eq}, we have \label{eqforv} \begin{gathered} -(r^{N-1}|u'|^{p-2}v')' = p^* \lambda r^{N-1} \partial_2 f(u) v, \quad 00 be the smallest positive zero of v. Without loss of generality, we can suppose v>0 on (0,r_1). If r_1<1, we have u(r_1)v'(r_1)<0. However by (H3), \begin{equation*}\label{signofv} r_1^{N-1}|u'(r_1)|^{p-2}u(r_1)v'(r_1)= \lambda\int_0^{r_1} s^{N-1}[f(u)-p^*\partial_2f(u)u]v\,\mathrm{d} s \geqslant 0, \end{equation*} a contradiction. If r_1=1, (H3) and Proposition~\ref{pos.prop} imply$$ 0=|u'(1)|^{p-2}u(1)v'(1)= \lambda \int_0^{1} s^{N-1}[f(u)-p^*\partial_2f(u)u]v\,\mathrm{d} s > 0, $$again a contradiction. Hence, v\not\equiv0 is impossible and so N(D_u F(\lambda,u))=\{0\}. \end{proof} By Remark~\ref{lambounds.rem}, Theorem~\ref{loc1.thm} and Lemma~\ref{IFT.lem}, the implicit function theorem yields a maximal open interval (\lambda_0,\widetilde\lambda) with \lambda_0<\widetilde\lambda\leqslant\lambda_\infty and a mapping u \in C^1((\lambda_0,\widetilde\lambda),Y) such that (\lambda,u(\lambda))\in\mathcal{S} for all \lambda\in(\lambda_0,\widetilde\lambda), and \lim_{\lambda\to\lambda_0}u(\lambda)=0. Let us show that \widetilde\lambda=\lambda_\infty. Suppose by contradiction that \lambda_0<\widetilde\lambda<\lambda_\infty\leqslant\infty, and consider a sequence \lambda_n\to\widetilde\lambda. If |u(\lambda_n)|_0 is unbounded, it follows by Lemma~\ref{asympt.lem} that \widetilde\lambda=\lambda_\infty and we are done. On the other hand, if |u(\lambda_n)|_0 is bounded, a compactness argument similar to that yielding the convergence of \{v_n\} in the proof of Lemma~\ref{asympt.lem} shows that there exists \widetilde u\in Y such that u(\lambda_n)\to\widetilde u (up to a subsequence), and \begin{equation*} \widetilde u = S_p(\widetilde\lambda f(\widetilde u)). \end{equation*} Note that, by Lemma~\ref{asympt.lem}, we cannot have u\equiv0. Hence, (\widetilde\lambda,\widetilde u)\in\mathcal{S}, F(\widetilde\lambda,\widetilde u)=0, and so by Lemma~\ref{IFT.lem} and the implicit function theorem, we can extend the curve u(\lambda) through the point (\widetilde\lambda,\widetilde u), contradicting the maximality of \widetilde\lambda. Therefore, \widetilde\lambda=\lambda_\infty, and we have a solution curve$$ \mathcal{S}_0:=\{(\lambda,u(\lambda)):\lambda\in(\lambda_0,\lambda_\infty)\} \subset\mathcal{S}. We next prove that \lim_{\lambda\to\lambda_\infty}|u(\lambda)|_0=\infty. In the case where (H6)(a) holds, this readily follows by the above argument for if |u(\lambda)|_0 were bounded as \lambda\to\lambda_\infty<\infty, we could continue the solution curve beyond \lambda=\lambda_\infty. In case (H6)(b) holds, the result follows from \begin{lemma}\label{blowup.lem} Suppose that {\rm (H6)(b)} holds, and consider (\lambda_n,u_n)\in\mathcal{S} with \lambda_n\to\infty. Then |u_n|_0\to\infty. \end{lemma} \begin{proof} By contradiction, suppose there exists a constant R>0 and a subsequence, still denoted by (\lambda_n,u_n), such that |u_n|_0\leqslant R for all n. Then, by (H2) and (H3), \begin{align*} |u_n|_0=u_n(0) &= \int_0^1\phi_{p'}\Big(\int_0^s\lambda_n f(t,u_n)\,\mathrm{d} t \Big)\,\mathrm{d} s\\ &\geqslant \lambda_n^{p^*} \int_0^{1/2} \phi_{p'} \Big(\int_0^s g(t,u_n)u_n(t)^{p-1}\,\mathrm{d} t \Big)\,\mathrm{d} s\\ &\geqslant \lambda_n^{p^*} u_n(1/2) \int_0^{1/2} \phi_{p'} \Big(\int_0^s g(t,R)\,\mathrm{d} t \Big)\,\mathrm{d} s\\ &\geqslant \lambda_n^{p^*} C|u_n|_0, \end{align*} where the last inequality follows from the concavity of the solutions u_n on [0,1]. Hence \lambda_n^{p^*}\leqslant C^{-1}<\infty, a contradiction. \end{proof} We still need to prove the uniqueness statement of Theorem~\ref{main.thm}, that is, \mathcal{S}_0=\mathcal{S}. Suppose instead that there exists (\bar\lambda,\bar u) \in \mathcal{S}\setminus\mathcal{S}_0, and let \mathcal{S}_1 be the connected subset of \mathcal{S} such that (\bar\lambda,\bar u) \in \mathcal{S}_1. It follows by Lemma~\ref{IFT.lem} that \mathcal{S}_1 is a smooth curve, parametrized by \lambda in a maximal interval I_1. In fact, the previous arguments imply that I_1=(\lambda_0,\lambda_\infty). Let us denote by u_1:(\lambda_0,\lambda_\infty)\to Y the parametrization of \mathcal{S}_1 and consider a sequence \lambda_n\to\lambda_0. Since |u_1(\lambda_n)|_0 is bounded by Lemma~\ref{asympt.lem}, it follows that there exists u_0 \in Y such that u(\lambda_n)\to u_0 in Y as n\to\infty. Then by continuity, we have \begin{equation*} u_0=S_p(\lambda_0f(u_0)). \end{equation*} Since u_1(\lambda_n)\geqslant0 for all n, it follows that u_0\geqslant0. We will show that, in fact, u_0\equiv0. Hence we will have (\lambda_n,u_1(\lambda_n))\to (\lambda_0,0) in Y and, by the characterization \eqref{loc1.eq} in Theorem~\ref{loc1.thm}, \mathcal{S}_1=\mathcal{S}_0. If u_0\not\equiv0, we set w_0=u_0/|u_0|_0. Then w_0\geqslant0 and satisfies $$\label{final.eq} w_0=S_p(\lambda_0g(u_0)\phi_p(w_0)).$$ Having in mind (H3) and \eqref{boundsforg}, it follows from the comparison theorem of \cite[Sec.~4]{w} applied to \eqref{final.eq} and \eqref{eigen} with f_0 that we must have w_0\equiv0. This contradiction finishes the proof of Theorem~\ref{main.thm}. \qed \subsection{Proof of Theorem~\ref{main2.thm}} We start with the analogue of Lemma~\ref{IFT.lem} under hypothesis (H'). \begin{lemma}\label{IFT'.lem} The function F \in C([0,\infty)\times Y, Y) defined in \eqref{dirop.eq} is continuously differentiable in a neighbourhood of any point (\lambda,u) \in \mathcal{S}^\pm, with D_u F(\lambda,u) v = v - \lambda DS_p(\lambda f(u)) \partial_2 f(u) v, \quad v\in Y. $$Furthermore, for any (\lambda,u) \in \mathcal{S}^\pm, D_u F(\lambda,u):Y\to Y is an isomorphism. \end{lemma} \begin{proof} The proof is almost identical to that of Lemma~\ref{IFT.lem}, so we only indicate the minor modifications. The differentiability part follows as in Lemma~\ref{IFT.lem}, using Theorem~\ref{Sp.thm}(ii), and Proposition~\ref{pos'.prop} instead of Proposition~\ref{pos.prop}. The non-singularity of D_u F(\lambda,u):Y\to Y follows in the same way if (\lambda,u)\in\mathcal{S}^+. For (\lambda,u)\in\mathcal{S}^-, we proceed in a similar manner, considering v\in N(D_u F(\lambda,u)). Then the identity \eqref{lagrange} still holds, and we suppose by contradiction that v>0 on a maximal interval (0,r_2), with v(r_2)=0 and v'(r_2)<0. If r_2<1, we have u(r_2)v'(r_2)>0 while (H3') implies \begin{equation*} r_2^{N-1}|u'(r_2)|^{p-2}u(r_2)v'(r_2)= \lambda\int_0^{r_2} s^{N-1}[f(u)-p^*\partial_2f(u)u]v\,\mathrm{d} s \leqslant 0, \end{equation*} a contradiction. On the other hand, if r_2=1, it follows from (H3') and Proposition~\ref{pos'.prop} that$$ 0=|u'(1)|^{p-2}u(1)v'(1)= \lambda \int_0^{1} s^{N-1}[f(u)-p^*\partial_2f(u)u]v\,\mathrm{d} s < 0, $$another contradiction. Hence v\equiv0 and N(D_u F(\lambda,u))=\{0\}. \end{proof} Now, similarly to the proof of Theorem~\ref{main.thm}, using Remark~\ref{lambounds'.rem}, Theorem~\ref{loc2.thm} and Lemma~\ref{IFT'.lem}, the implicit function theorem yields maximal open intervals (\lambda_0,\widetilde\lambda_\pm) with \lambda_0<\widetilde\lambda_\pm\leqslant\lambda_\infty and two solution curves u_\pm \in C^1((\lambda_0,\widetilde\lambda_\pm),Y). It follows as in the proof of Theorem~\ref{main.thm} that \widetilde\lambda_\pm=\lambda_\infty, so we get two global solution curves$$ \mathcal{S}_0^\pm=\{(\lambda,u_\pm(\lambda)): \lambda\in(\lambda_0,\lambda_\infty)\}\subset\mathcal{S}.  Furthermore, $\lim_{\lambda\to\lambda_\infty}|u(\lambda)|_0=\infty$ follows as in the proof of Theorem~\ref{main.thm}, using the version of Lemma~\ref{blowup.lem} holding under hypothesis (H'): \begin{lemma}\label{blowu'p.lem} Suppose that {\rm (H6)(b)} holds, and consider $(\lambda_n,u_n)\in\mathcal{S}^\pm$ such that $\lambda_n\to\infty$. Then $|u_n|_0\to\infty$. \end{lemma} \begin{proof} The proof is the same as that of Lemma~\ref{blowup.lem} when $(\lambda_n,u_n)\in\mathcal{S}^+$. For $(\lambda_n,u_n)\in\mathcal{S}^-$, if $|u_n|_0\leqslant R$, it follows by (H2') and (H3') that \begin{align*} |u_n|_0=-u_n(0)&=-\int_0^1\phi_{p'}\left(\int_0^s\lambda_n f(t,u_n)\,\mathrm{d} t \right)\,\mathrm{d} s\\ &\geqslant \lambda_n^{p^*} \int_0^{1/2} \phi_{p'} \Big(\int_0^s g(t,u_n)|u_n(t)|^{p-1}\,\mathrm{d} t \Big)\,\mathrm{d} s\\ &\geqslant \lambda_n^{p^*} |u_n(1/2)| \int_0^{1/2} \phi_{p'} \Big(\int_0^s g(t,R)\,\mathrm{d} t \Big)\,\mathrm{d} s\\ &\geqslant \lambda_n^{p^*} C|u_n|_0, \end{align*} showing that the sequence $\{\lambda_n\}$ must be bounded, a contradiction. \end{proof} Using the characterization \eqref{loc2.eq} in Theorem~\ref{loc2.thm}, it follows similarly to the last part of the proof of Theorem~\ref{main.thm} that $\mathcal{S}=\mathcal{S}_0^-\cup\mathcal{S}_0^+$. Hence, to finish the proof of Theorem~\ref{main2.thm}, we only need to prove that $\mathcal{S}_0^-\subset\mathcal{S}^-$ and $\mathcal{S}_0^+\subset\mathcal{S}^+$. Let $\mathcal{C}_0^\pm:=\mathcal{S}_0^\pm\cap\mathcal{S}^\pm$. First, we know from Theorem~\ref{loc2.thm} that, for $\lambda$ close to $\lambda_0$, $(\lambda,u_\pm(\lambda))\in\mathcal{S}^\pm$, so that $\mathcal{C}_0^\pm\neq\emptyset$. Thus, we need only show that $\mathcal{C}_0^\pm$ is both open and closed in $\mathcal{S}_0^\pm$, for the product topology inherited from $\mathbb{R}\times Y$. We will only consider $\mathcal{C}_0^+$, the proof for $\mathcal{C}_0^-$ is similar. For $(\lambda,u)\in \mathcal{C}_0^+$, it follows from Proposition~\ref{pos'.prop} that $u>0$ on $[0,1)$ (with $u(1)=0$). If $(\mu,v)\in \mathcal{S}_0^+$ with $|\mu-\lambda|+|v-u|_0$ small enough, we will have $\lambda\in(\lambda_0,\lambda_\infty)$ and $0\not\equiv v\geqslant0$ on $[0,1]$, hence $(\mu,v)\in\mathcal{C}_0^+$. This proves that $\mathcal{C}_0^+$ is open in $\mathcal{S}_0^+$. Now consider a sequence $\{(\lambda_n,u_n)\}\subset\mathcal{C}_0^+$ and suppose there exists $(\lambda,u)\in \mathcal{S}_0^+$ such that $(\lambda_n,u_n) \to (\lambda,u)$. By continuity, $F(\lambda,u)=0$, and we have $u\geqslant0$. Since $\lambda>\lambda_0$, it follows from Lemma~\ref{asympt'.lem} that $u\not\equiv0$. Hence, $(\lambda,u)\in\mathcal{S}_0^+$ and $\mathcal{C}_0^+$ is closed in $\mathcal{S}_0^+$. Thus, $\mathcal{C}_0^+=\mathcal{S}_0^+$, and it follows in a similar way that $\mathcal{C}_0^-=\mathcal{S}_0^-$, showing that $\mathcal{S}_0^\pm\subset\mathcal{S}^\pm$, and so actually $\mathcal{S}_0^\pm=\mathcal{S}^\pm$. This completes the proof of Theorem~\ref{main2.thm}. \qed \subsection*{Acknowledgements} We are grateful to the anonymous referee for drawing our attention to the reference \cite{bro}. \begin{thebibliography}{99} \bibitem{br} P. A. Binding, B. P. Rynne; The spectrum of the periodic $p$-Laplacian, {\it J. 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