\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \documentclass[twoside]{article} \usepackage{amssymb, amsmath} \pagestyle{myheadings} \markboth{\hfil Global bifurcation result for the p-biharmonic operator \hfil EJDE--2001/48} {EJDE--2001/48\hfil Pavel Dr\'abek \& Mitsuharu \^Otani \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2001}(2001), No. 48, pp. 1--19. \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} \\ % Global bifurcation result for the p-biharmonic operator % \thanks{ {\em Mathematics Subject Classifications:} 35P30, 34C23. \hfil\break\indent {\em Key words:} p-biharmonic operator, principal eigenvalue, global bifurcation. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Submitted February 9, 2001. Published July 3, 2001.} } \date{} % \author{Pavel Dr\'abek \& Mitsuharu \^Otani} \maketitle \begin{abstract} We prove that the nonlinear eigenvalue problem for the p-biharmonic operator with $p > 1$, and $\Omega$ a bounded domain in $\mathbb{R}^N$ with smooth boundary, has principal positive eigenvalue $\lambda_1$ which is simple and isolated. The corresponding eigenfunction is positive in $\Omega$ and satisfies $\frac{\partial u}{\partial n} < 0$ on $\partial \Omega$, $\Delta u_1 < 0$ in $\Omega$. We also prove that $(\lambda_1,0)$ is the point of global bifurcation for associated nonhomogeneous problem. In the case $N=1$ we give a description of all eigenvalues and associated eigenfunctions. Every such an eigenvalue is then the point of global bifurcation. \end{abstract} \def\Deg{\mathop{\rm Deg}} \def\sgn{\mathop{\rm sgn}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 \section{Introduction} Let $\Omega \subset \mathbb{R}^N$ be a bounded domain with smooth boundary $\partial \Omega$. For $p \in (1, + \infty)$ consider the nonlinear eigenvalue problem $$\begin{gathered} \Delta (|\Delta u|^{p-2} \Delta u) = \lambda |u|^{p-2}u \quad\mbox{in } \Omega \\ u = \Delta u = 0 \quad \mbox{on } \partial \Omega \end{gathered} \label{ENp}$$ In this paper we prove that (\ref{ENp}) has a {\it principal positive eigenvalue} $\lambda_1 = \lambda_1 (p)$ which is {\it simple} and {\it isolated}. Moreover, we prove that there exists {\it strictly positive eigenfunction} $u_1 = u_1 (p)$ in $\Omega$ associated with $\lambda_1 (p)$ and satisfying $\frac{\partial u_1}{\partial n} < 0$ on $\partial \Omega$. We also study the dependence of $\lambda_1 (p)$ on $p$ and show that $p \mapsto \lambda_1 (p)$ is a {\it continuous} function in $(1, + \infty)$. Making use of this result we prove that $\lambda_1 (p)$ is a {\it bifurcation point} of $$\begin{gathered} \Delta (|\Delta u|^{p-2} \Delta u) = \lambda |u|^{p-2}u + g (x, \lambda, u) \quad\mbox{in } \Omega \\ u = \Delta u = 0 \quad\mbox{on } \partial \Omega , \end{gathered}\label{BPNp}$$ from which a global continuum of nontrivial solutions emanates. In {\it one dimensional case} $(N = 1, \Omega = (0,1))$ we obtain a {\it complete characterization} of the {\it spectrum} of the eigenvalue problem $$\begin{gathered} (|u''|^{p-2} u'')'' = \lambda |u|^{p-2}u \quad\mbox{in } (0,1)\\ u (0) = u''(0) = u(1) = u''(1) = 0. \end{gathered} \label{E1p}$$ We prove that the spectrum of (\ref{E1p}) consists of a sequence of simple eigenvalues $0 < \lambda_1 < \ldots < \lambda_n < \ldots \to +\infty$. The eigenfunction $u_n$ associated with $\lambda_n (n \geq 2)$ has precisely $n$ bumps in $(0,1)$ and it is reproduced from $u_1$ by using the symmetry of (\ref{E1p}). As a simple consequence we then obtain that any $\lambda_n$ is a global bifurcation point of $$\begin{gathered} (|u''|^{p-2} u'')'' = \lambda |u|^{p-2}u + g (t , \lambda, u) \quad\mbox{in }(0,1)\\ u (0) = u''(0) = u(1) = u''(1) = 0. \end{gathered} \label{BP1p}$$ Our main results are stated in the following theorems. \begin{theorem}\label{th1.1} The problem (\ref{ENp}) has the least positive eigenvalue $\lambda_1 (p)$ which is simple and isolated in the sense that the set of all solutions of (\ref{ENp}) with $\lambda = \lambda_1 (p)$ forms a one dimensional linear space spanned by a positive eigenfunction $u_1(p)$ associated with $\lambda_1(p)$ such that $\Delta u_1(p) < 0$ in $\Omega$ and $\frac{\partial u_1(p)}{\partial n} < 0 \mbox{ on } \partial \Omega$ and that there exists a positive number $\delta$ so that $(\lambda_1(p), \lambda_1(p) + \delta)$ does not contain any eigenvalues of ${\rm (E_N)_p.}$ Moreover, (\ref{ENp}) has a positive solution if and only if $\lambda = \lambda_1$ and the function $p \mapsto \lambda_1 (p)$ is continuous. \end{theorem} \begin{theorem}\label{th1.2} Let $p>1$ be fixed and the function $g = g (x,\lambda, s)$, $g (x,\lambda,0)= 0$, represents higher order terms in (\ref{BPNp}) (see Section 4 for precise assumptions). Then there exists a continuum of nontrivial solutions $(\lambda, u)$ of (\ref{BPNp}) bifurcating from $(\lambda_1 (p), 0)$ which is either unbounded or meets the point $(\lambda_e (p), 0)$, where $\lambda_e (p) > \lambda_1 (p)$ is some eigenvalue of (\ref{ENp}). \end{theorem} \begin{theorem}\label{th1.3} The set of all eigenvalues of (\ref{E1p}) is formed by a sequence $$0 < \lambda_1 (p) < \lambda_2 (p) < \ldots < \lambda_n (p) < \ldots \to + \infty .$$ For any $n = 1,2, \ldots,$ the function $p \mapsto \lambda_n (p)$ is continuous. Every $\lambda_n (p)$ is simple and the corresponding one dimensional space of solutions of (\ref{E1p}) with $\lambda = \lambda_n (p)$ is spanned by a function having precisely $n$ bumps in $(0,1)$. Each $n$-bump solution is constructed by the reflection and compression of the eigenfunction $u_1(p)$ associated with $\lambda_1 (p)$. \end{theorem} \begin{theorem}\label{th1.4} Let $p > 1$ be fixed and $g = g (t, \lambda, s), g (t, \lambda, 0)= 0$, represents higher order terms in (\ref{BP1p}) (see Section 5 for precise assumptions). Then for every $n = 1,2, \ldots$ there exists a continuum of nontrivial solutions $(\lambda, u)$ of (\ref{BP1p}) bifurcating from $(\lambda_n (p), 0)$ which is either unbounded or meets the point $(\lambda_k (p), 0)$, with $k \neq n$. \end{theorem} The paper is organized as follows. In Section 2 we define the notion of the solution, and prepare some auxiliary results. Section 3 contains the proof of Theorem \ref{th1.1}. The essential part of it relies on the abstract result of Idogawa and \^Otani \cite{IO} and the verification of its assumptions. In Section 4 we prove the bifurcation result stated in Theorem \ref{th1.2} using the degree argument and the well-known result of Rabinowitz [R]. The last Section 5 is devoted to the one dimensional case and Theorems \ref{th1.3}, \ref{th1.4} are proved there. \section{Auxiliaries} For $p>1$ we define the function $\psi_p : \mathbb{R} \to \mathbb{R}$ by $\psi_p (s)= |s|^{p-2} s, s \neq 0$ and $\psi_p (0) = 0$. Denoting $p' = \frac{p}{p-1}$ we immediately obtain that $z = \psi_p (s)$ if and only if $s = \psi_{p'} (z)$. The eigenvalue problem (\ref{ENp}) can be thus written in the form $$\label{2.1} \begin{gathered} \Delta \psi_p (\Delta u) = \lambda \psi_p (u) \quad\mbox{in } \Omega\\ u = \Delta u = 0 \quad\mbox{on } \partial \Omega . \end{gathered}$$ Before we define the weak solution to (\ref{2.1}) we recall some properties of the Dirichlet problem for Poisson equation: $$\label{2.2} \begin{gathered} - \Delta w = f \quad\mbox{in } \Omega \\ w = 0 \quad\mbox{on } \partial \Omega . \end{gathered}$$ It is well known that (\ref{2.2}) is uniquely solvable in $L^p(\Omega)$ for any $p \in (1, \infty )$ and that the linear solution operator $\Lambda : L^p (\Omega) \to W^{2,p} (\Omega) \cap W_0^{1,p} (\Omega)$, $\Lambda f =w$, has the properties stated in the following lemma, (see, e.g., \cite{GT}). \begin{lemma}\label{lem2.1} \begin{itemize} \item[(i)] (Continuity) There exists a constant $c_p > 0$ such that $$\| \Lambda f \|_{W^{2,p}} \leq c_p \|f\|_{L^p}$$ holds for all $p \in (1, \infty)$ and $f \in L^p (\Omega)$. \item[(ii)] (Continuity) Given $k \geq 1, k \in \mathbb{N}$, there exists a constant $c_{p,k} > 0$ such that $$\| \Lambda f\|_{W^{k+2,p}} \leq c_{p,k} \|f\|_{W^{k,p}}$$ holds for all $p \in (1, \infty)$ and $f \in W^{k,p} (\Omega)$. \item[(iii)] (Symmetry) The following identity $$\int_{\Omega} \Lambda u \cdot v dx = \int_{\Omega} u \cdot \Lambda v dx$$ holds for all $u \in L^p(\Omega)$ and $v \in L^{p'} (\Omega)$ with $p \in (1, \infty)$. \item[(iv)] (Regularity) Given $f \in L^{\infty} (\Omega)$, we have $\Lambda f \in C^{1, \alpha} (\bar{\Omega})$ for all $\alpha \in (0,1) ;$ moreover, there exist $c_{\alpha} > 0$ such that $$\| \Lambda f\|_{C^{1,\alpha}} \leq c_{\alpha} \|f\|_{L^{\infty}}.$$ \item[(v)] (Regularity and Hopf-type maximum principle) Let $f \in C (\bar{\Omega})$ and $f \geq 0$, then $w = \Lambda f \in C^{1, \alpha} (\bar{\Omega})$, for all $\alpha \in (0,1)$ and $w$ satisfies: $w > 0$ in $\Omega, \frac{\partial w}{\partial n} < 0$ on $\partial \Omega$. \item[(vi)] (Order preserving property) Given $f, g \in L^p (\Omega), f \leq g$ in $\Omega$, we have $\Lambda f < \Lambda g$ in $\Omega$. \end{itemize} \end{lemma} Let us denote $v : = - \Delta u$ in (\ref{ENp}). Then the problem (\ref{ENp}) can be restated as an operator equation $$\label{2.3} \psi_p (v) = \lambda \Lambda \psi_p (\Lambda v) \quad \mbox{in } \Omega$$ or as $$\label{2.4} v = \lambda^{\frac{1}{(p-1)}} \psi_{p'} (\Lambda \psi_p (\Lambda v)) \quad\mbox{in }\Omega.$$ Indeed, let us assume that $v \in L^p (\Omega)$ solves (\ref{2.3}). Then from Lemma \ref{lem2.1} (i) and the properties of the Nemytskii operator induced by $\psi_p$ we obtain: \begin{eqnarray*} &&u = \Lambda v \in W^{2,p} (\Omega) \cap W^{1,p}_{0} (\Omega) \Rightarrow \psi_p (\Lambda v) \in L^{p'} (\Omega) \Rightarrow \\ && \Rightarrow \Lambda \psi_p (\Lambda v) \in W^{2,p'} (\Omega) \cap W^{1,p'}_{0} (\Omega) \Rightarrow\\ && \Rightarrow \psi_p (v) \in W^{2,p'} (\Omega) \cap W^{1, p'}_{0} (\Omega)\Rightarrow\\ &&\Rightarrow - \Delta \psi_p (- \Delta u) = \lambda \psi_p (u) \mbox{ holds in } L^{p'}(\Omega). \end{eqnarray*} This enables us to give the following definition of the solution of (\ref{ENp}). \begin{definition}\label{def2.2} \rm The function $u \in W^{2,p} (\Omega) \cap W^{1,p}_0 (\Omega)$ is called {\it a solution} of (\ref{ENp}) if $v = - \Delta u$ solves (\ref{2.3}) in $L^{p'} (\Omega)$. The parameter $\lambda_e$ is called an {\it eigenvalue} of (\ref{ENp}) if there is a nonzero solution $u_e$ of (\ref{ENp}) with $\lambda = \lambda_e$. The function $u_e$ is then called the {\it eigenfunction} associated with the eigenvalue $\lambda_e$. \end{definition} \begin{lemma}\label{lem2.3} (Duality). Let $\lambda_e = \lambda_e (p) \neq 0$ be the eigenvalue of $(E_N)_p$ and $u_e (p)$ be the eigenfunction associated with $\lambda_e$. Define $\lambda_e^{(p')}$ and $u_e (p')$ by \break $\lambda_{e}^{1/p} (p) = \lambda^{1/p'}_{e} (p')$ and $u_e (p') = \lambda^{-1}_e (p) \psi_p (\Delta u_e (p))$. Then $\lambda_e (p')$ becomes an eigenvalue of $(E_N)_{p'}$ with $p'= \frac{p}{p - 1}$ and $u_e (p')$ gives the eigenfunction associated with $\lambda_e (p')$. \end{lemma} \paragraph{Proof.} We have $$\label{2.5} \begin{gathered} \Delta \psi_p (\Delta u_e (p)) = \lambda_e (p) \psi_p (u_e(p)) \quad\mbox{in }\Omega \\ u_e (p) = \Delta u_e (p) = 0 \quad\mbox{on }\partial \Omega . \end{gathered}$$ Let $w_p: = \psi_p (\Delta u_e (p))$, then $w_p \in W^{2,p'} (\Omega)\cap W^{1,p'}_0 (\Omega)$. It is easy to see that to solve (\ref{2.5}) is nothing but to find $(u_e (p), w_p)$ satisfying the system $$\label{2.5p} \begin{gathered} \Delta w_p = \lambda_e (p) \psi_p (u_e(p)) \\ \Delta u_e (p) = \psi_{p'} (w_p). \end{gathered}$$ Since $u_e (p') = \frac{1}{\lambda_e (p)} w_p \in W^{2,p'} (\Omega) \cap W^{1,p'}_0 (\Omega)$ satisfies $\psi_{p'} (u_e (p')) =\\ \lambda_e (p)^{1 -p'} \psi_{p'} (w_p) = \lambda_e (p')^{-1} \psi_{p'} (w_p)$, we easily find that $(u_e(p'), w_{p'})$ with $w_{p'} = u_e(p)$ solves (\ref{2.5p}) with $p=p'$. \begin{remark}\label{rem2.4}\rm The duality proved in the previous lemma enables us to deduce several properties of (\ref{ENp}) for $p > 2$ from those for $p \in (1,2)$ and vice versa. \end{remark} The following technical lemma will be useful for the verification of certain abstract assumptions in the next section. \begin{lemma}\label{lem2.5} Let $A,B,C$ and $p$ be real numbers satisfying $A \geq 0, B \geq 0, C \geq \max \{B - A, 0 \}$ and $p> 1$. Then $$\label{2.6} |A + C|^p + |B - C|^p \geq A^p + B^p.$$ \end{lemma} \paragraph{Proof.} If $C = 0$ (i.e, $B \leq A$), then (\ref{2.6}) is trivial. So it suffices to show (\ref{2.6}) when $B \geq A$. Due to the strict convexity of the function $s \mapsto s^p$, in $(0, + \infty)$ we have \begin{gather*} |A + C|^p \geq B^p + p B^{p-1} [C - (B-A)],\\ | B - C|^p \geq A^p - p A^{p-1} [C - (B-A)]. \end{gather*} Adding these inequalities, we derive the assertion. \quad$\Box$ \section{Eigenvalue problem} Let us define convex functionals $f^1_p, f^2_p: L^p (\Omega) \to \mathbb{R}$ as follows: $$f^1_p (v) = \frac{1}{p} \int_{\Omega} |v|^p dx,\; f^2_p (v) = \frac{1}{p} \int_{\Omega} |\Lambda v|^p dx.$$ Then it is clear that $f^1_p$ and $f^2_p$ are Fr\' echet differentiable in $L^p(\Omega)$. Since for every Fr\' echet differentiable convex functional $f$, its subdifferential $\partial f$ coincides with its Fr\'echet derivative $f'$, we get that (\ref{2.3}) is equivalent to $$\label{3.1} \partial f_p^1 (v) = \lambda \partial f_p^2 (v) \mbox{ in } L^{p'} (\Omega),$$ where $\partial f^i_p$ are the subdifferentials of $f^i_p, (i = 1,2)$. We are going to verify the hypotheses $(A0), (A0)', (\ref{A1}) - (\ref{A6})$ of \cite{IO} with $A = \partial f^1_p$, $B = \partial f^2_p$ and $V= L^p (\Omega)$. The assumptions (\ref{A1}) (i)--(iii), (\ref{A2}) (i)--(iii), (\ref{A3}), (\ref{A4}) (i) and (\ref{A5}) are clearly satisfied. Concerning (\ref{A4}) (ii) we should verify that $$\label{3.2} f^2_p (\max\{u,w \}) + f_p^2 (\min \{u, w \}) \geq f^2_p (u) + f_p^2 (w)$$ for any $u,w \in L^p (\Omega)$ satisfying $u \geq 0$ and $w \geq 0$ a.e. in $\Omega$. We have $\max \{u, w \} = u + (w - u)^+$ and $\min \{u, w \} = w - (w - u)^+.$ By Lemma \ref{lem2.1} (vi), the inequality $w - u \leq (w -u)^+$ implies $\Lambda (w -u)^+ \geq \Lambda (w -u) = \Lambda w - \Lambda u$. Hence Lemma \ref{lem2.5} with $A = \Lambda u, B = \Lambda w$ and $C = \Lambda (w -u)^+$ gives $$\label{3.3} \int_{\Omega} |\Lambda u + \Lambda (w - u)^+|^p dx + \int_{\Omega} |\Lambda w - \Lambda (w - u)^+|^p dx \geq \int_{\Omega} |\Lambda u|^p dx + \int_{\Omega} |\Lambda w|^p dx.$$ Then (\ref{3.3}) implies (\ref{3.2}). The assumption (\ref{A6}) is a consequence of Lemma \ref{lem2.1} (vi). Hence it remains to verify (A0) and (A0)'. \begin{lemma}\label{lem3.1} Let $v \in L^p(\Omega)$ solve (\ref{2.3}) in $L^{p'} (\Omega)$. Then $v \in C (\bar{\Omega})$. \end{lemma} \paragraph{Proof.} The main part of the proof is to show the following fact:\\ Suppose, that $v \in L^{p_0} (\Omega)$, then we find that \begin{enumerate} \item[(i)] $v \in L^{p_1} (\Omega)$ with $\frac{1}{p_1} = \frac{1}{p_0} - \frac{ p'}{N}$ if $p_0 < \frac{N}{2p'}$ \item[(ii)] $v \in C (\bar{\Omega})$ if $p_0 > \frac{N}{2p'}$, $p' =\frac{p}{p-1}$. \end{enumerate} Let $v \in L^{p_0} (\Omega)$, and $p_0 < \frac{N}{2p}$, then $\Lambda v \in W^{2,p_0} (\Omega)$ by Lemma \ref{lem2.1}(i). Then, by Sobolev's embedding theorem and the property of the Nemytskii operator: $r \mapsto \psi_p (r)$, we get $\Lambda v \in L^{r_0} (\Omega)$ and $\psi_p (\Lambda v) \in L^{\frac{r_0}{p-1}}$ with $r_0 = \frac{N p_0}{N - 2 p_0}$ Again, by Sobolev's embedding theorem and the property of the Nemytskii operator, we obtain $$\Lambda \psi_p (\Lambda v) \in W^{2,\frac{ro}{p-1}} (\Omega) \hookrightarrow L^{r_1} (\Omega)$$ and $$\psi_{p'} (\Lambda \psi_p (\Lambda v)) \in L^{\frac{r_1}{p' -1}} (\Omega) = L^{r_1(p - 1)} (\Omega)$$ with $r_1 = \frac{N r_0}{N (p - 1) - 2 r_0}$. Consequently, (\ref{2.4}) implies that $v \in L^{p_1} (\Omega)$ with $p_1 = r_1 (p-1)$, i.e., $\frac{1}{p_1} = \frac{1}{p_0} - \frac{2p'}{N}$, whence follows assertion (i). If $\frac{N}{2} < p_0$ it is obvious by Sobolev's embedding theorem that $v \in C (\bar \Omega)$. As for the case $\frac{N}{2p'} < p_0 < \frac{N}{2}$ (or $p_0 = \frac{N}{2}$), noting that $W^{2,\frac{r_0}{p-1}} (\Omega) \hookrightarrow C (\bar{\Omega})$ (or $W^{2, \frac{r}{p-1}} (\Omega) \hookrightarrow C (\bar{\Omega})$ for sufficiently large $r$) we easily see that $v \in C (\bar{\Omega})$. Then assertion ${\rm (ii)}$ is verified. Now take suitable $p_0 \in (1,p]$ and $k \in \mathbb{N}$ such that $$p_{k-1} < \frac{N}{2p'} < p_k \mbox{ with } \frac{1}{p_k}= \frac{1}{p_0} - \frac{2p'}{N} k.$$ Then applying assertion (i) with $p_0 = p_0, p_1 \ldots , p_{k-1}$, we deduce $v \in L^{p_k} (\Omega)$. Hence from assertion (ii), $v \in C (\bar{\Omega})$ follows. \quad$\Box$ \begin{remark}\label{rem3.2}\rm In particular, it follows from above proof that given bounded sequences $\{p_n\} \subset (1, \infty )$ and $\{ \lambda_n\} \subset (0, \infty )$, the sequence of elements $v_n$ solving (\ref{2.3}) with $\lambda =\lambda_n$ and $p = p_n$ which are normalized by $\|v_n\|_{L^q} = 1, q \in (1,\infty)$, we find a constant $c>0$ (independent of $n$) such that $$\|v_n\|_{L^{\infty}} \leq c.$$ By the same reason, if $\lambda_n \to \lambda_0$ and $v_0$ solves (\ref{2.3}) with $\lambda = \lambda_0, \|v_0\|_{L^q} = 1$, the proof of Lemma \ref{3.1} implies that $$\lim_{n \to \infty} \|v_n -v_0\|_{L^{\infty}} = 0.$$ \end{remark} \begin{lemma}\label{lem3.3} Let $p \geq 2$ and $v \in L^p (\Omega), v \geq 0$ a.e. in $\Omega$, and let $v$ solve (\ref{2.3}) in $L^{p'} (\Omega)$. Then $v \in C^1 (\Omega), v > 0$ everywhere in $\Omega$ and $\frac{\partial v}{\partial n} = - \infty$ on $\partial \Omega$. \end{lemma} \paragraph{Proof.} It follows from Lemma \ref{lem2.1} (v), Lemma \ref{lem3.1} and (\ref{2.3}) that $w : = \psi_p (v)$ satisfies $w \in C^{1, \alpha} (\bar{\Omega}), \alpha \in (0,1), w > 0$ in $\Omega$ and $\frac{\partial w}{\partial n} < 0$ on $\partial \Omega$. This fact assures that $v > 0$ in $\Omega$ and $(p-1) |v|^{p-2} \frac{\partial v}{\partial n} < 0$ on $\partial \Omega$. Then $\frac{\partial v}{\partial n} = -\infty$ follows from the fact that $v = 0$ on $\partial \Omega$. \quad$\Box$ \medskip For $p \geq 2$ the assumption (A0) now follows from Lemma \ref{lem3.3} while instead of (A0)' we obtain the following property - (A0)'': {\it Every positive solution $v$ of (\ref{3.1}) satisfies $v \in C^1 (\Omega), v = 0$ on $\partial \Omega$ and $\frac{\partial v}{\partial n} = - \infty$ on $\partial \Omega.$} \\ It is easy to see that the results of \cite{IO} remain true even if (A0)' is substituted by (A0)''. Applying now the results of \cite{IO} we deduce that, for $p \geq 2$, $$0 < \lambda_1 (p) : = \Bigg(\sup\limits_{v \in L^p (\Omega) \,\; v\neq 0} \frac{f^2_p (v)}{f^1_p (v)}\Bigg)^{-1},$$ is the least simple eigenvalue of (\ref{3.1}) with associated positive eigenfunction $v_1 (p), \| v_1 (p) \|_{L^p} = 1$ and (\ref{3.1}) has a positive solution if and only if $\lambda = \lambda_1 (p)$. The assertion for $p \in (1,2)$ now follows from Lemma \ref{lem2.3} and Remark \ref{rem2.4}. As a consequence of this fact we find that $u_1 (p) = \Lambda v_1 (p)$ is the corresponding first eigenfunction of ${\rm(E_N)_p}$ satisfying $u_1(p) > 0$ in $\Omega, \Delta u_1 (p) < 0$ in $\Omega$ and $\frac{\partial u_1 (p)}{\partial n} < 0$ on $\partial \Omega$ due to Lemma \ref{lem2.1} (vi). Moreover, if u is another positive solution of ${\rm(E_N)_p}$ then $v = -\Delta u > 0$ solves (\ref{2.3}) in $L^{p'}(\Omega)$. Therefore (\ref{2.4}) holds with $\Lambda v = u$. Hence according to the above mentioned argument, it holds that $\lambda = \lambda_1 (p)$ and $v = v_1 (p)$, i.e. $u = u_1(p)$. \begin{lemma}\label{lem3.4} $\lambda_1 (p)$ is isolated, i.e. there is $\delta > 0$ such that the interval $(\lambda_1(p), \lambda_1$\\ $(p) + \delta)$ does not contain any eigenvalue of (\ref{3.1}). \end{lemma} \paragraph{Proof.} Assume the contrary, i.e., there are sequences $\{\lambda_n\} , \{v_n\}$ such that $\lambda_n \to \lambda_1 (p), \|v_n\|_{L^p} = 1$ and that $v_n$ solves (\ref{3.1}) with $\lambda = \lambda_n$. Then both $v_n$ and $\Lambda v_n$ must change sign in $\Omega$ and $$\lim_{n \to \infty} \|v_n -v_1 (p)\|_{L^{\infty}} = 0$$ according to Remark \ref{rem3.2}. But Lemma \ref{lem2.1} (iv) implies that $\Lambda v_n \to \Lambda v_1 (p)$ in $C^{1, \alpha} (\bar{\Omega})$ for some $\alpha \in (0,1)$ which leads to a contradiction with the fact that $\Lambda v_1 (p) > 0 \mbox{ in } \Omega$ and $\frac{\partial \Lambda v_1 (p)}{\partial n} < 0 \mbox{ on } \partial \Omega$. \quad$\Box$\medskip It remains to show the continuity of $p \mapsto \lambda_1 (p)$. Let us note first that $$\lambda_1 (p) = \inf \frac{1}{f^2_p(v)},$$ where the infimum is taken over all $v \in L^p (\Omega), \|v\|_{L^p} = p$. It follows from Lemma \ref{lem2.1} (i) that $\lambda_1 (p)$ is bounded uniformly away from zero and infinity for any $p$ belonging to a compact subinterval of $(1,\infty)$. Let $p_n \to p \in (1,\infty)$. Then $\{\lambda_1 (p_n)\}$ is a bounded sequence. Denote by $v_1 (p_n)$ the positive eigenfunction associated with $\lambda_1 (p_n)$ and normalized by $$\label{3.4} \|v_1 (p_n)\|_{L^p} = p.$$ Extracting a suitable subsequence we can assume that $$\label{3.5} \lambda_1 (p_n) \to \lambda_0, v_1 (p_n) \rightharpoonup v_0 \mbox{ in } L^p (\Omega).$$ In particular, we derive from (\ref{3.5}) that $v_0 \geq 0$ a.e. in $\Omega$, and the compactness of $\Lambda$ (cf. Lemma \ref{lem2.1} (i)) yields $\Lambda v_1 (p_n) \to \Lambda v_0$ in $L^{p} (\Omega)$. Extracting again to a subsequence we get $$\label{3.6} \Lambda v_1 (p_n) \to \Lambda v_0 \mbox{ a.e in } \Omega.$$ It follows from Remark \ref{rem3.2} and Lemma \ref{lem2.1} (iv) that there is a constant $c > 0$ independent of $n$ such that $$\label{3.7} |\Lambda v_1 (p_n)| \leq c.$$ Hence it follows from (\ref{3.6}), (\ref{3.7}) and Lemma \ref{lem2.1} (iv) that $$\Lambda \psi_{p_n} (\Lambda v_1 (p_n)) \to \Lambda \psi_p (\Lambda v_0)\mbox{ a.e. in } \Omega, \mbox{ i.e., }$$ $$\label{3.8} \psi_{p'_n} (\Lambda \psi_{p_n} (\Lambda v_1 (p_n))) \to \psi_{p'} (\Lambda (\psi_p (\Lambda v_0))) \mbox{ a.e. in } \Omega.$$ Now taking arbitrary $\varphi \in L^{p'} (\Omega)$, it follows from (\ref{3.4}), (\ref{3.5}), (\ref{3.7}), (\ref{3.8}), Lemma \ref{lem2.1} (iv) and the Lebesgue dominated convergence theorem that $$\label{3.9} \int_{\Omega} \lambda_1^{\frac{1}{p_n -1}} (p_n) \psi_{p'_n} (\Lambda \psi_{p_n} (\Lambda v_1 (p_n))) \varphi dx \to \int_{\Omega} \lambda_0^{\frac{1}{p-1}} \psi_{p'} (\Lambda \psi_p (\Lambda v_0)) \varphi dx.$$ It also follows from (\ref{3.5}) that $$\label{3.10} \int_{\Omega} v_1 (p_n) \varphi dx \to \int_{\Omega} v_0 \varphi dx.$$ So it follows from (\ref{2.4}), (\ref{3.9}) and (\ref{3.10}) that $$\label{3.11} v_0 = \lambda_0^{\frac{1}{p-1}} \psi_{p'} (\Lambda \psi_p (\Lambda v_0)).$$ On the other hand (\ref{3.6}), (\ref{3.7}) the definition of $\lambda_1$ and the Lebesgue dominated convergence theorem imply $$1 = \lim_{n \to \infty} \lambda_1 (p_n) \int_{\Omega} |\Lambda v_1 (p_n)|^{p_n} dx = \lambda_0 \int_{\Omega} |\Lambda v_0|^p dx,$$ i.e. $v_0 \not \equiv 0$. It follows from here and (\ref{3.11}) that $v_0$ is a positive solution of (\ref{2.3}) with $\lambda = \lambda_0$. According to the first part of Theorem \ref{th1.1} (cf.\cite{IO}) it must be $\lambda_0 = \lambda_1 (p), v_0 = v_1 (p)$. Since the above argument does not depend on the choice of subsequences, the continuity of the function $$p \mapsto \lambda_1 (p)$$ is proved. This also completes the proof of Theorem \ref{th1.1} \section{Global bifurcation result} For $p > 1$ set $X = L^p (\Omega)$. Then $X^* = L^{p'}(\Omega)$ and the Nemytskii operator $$\Psi_p : v \mapsto \psi_p (v)$$ is one to one mapping between $X$ and $X^*$. \begin{lemma}\label{lem4.1} $\Psi_p$ satisfies condition $(S_{+})$, i.e. $$\label{4.1} v_n \rightharpoonup v_0 \mbox{ weakly in } X.$$ and $$\label{4.2} \limsup_{n \to \infty} \int_{\Omega} \psi_p (v_n) (v_n - v_0) dx \leq 0$$ imply $v_n \to v_0 \mbox{ strongly in } X$. \end{lemma} \paragraph{Proof.} The monotonicity of $\psi_p$, (\ref{4.1}) and (\ref{4.2}) imply \begin{eqnarray*} 0 &\geq &\limsup_{n \to \infty} \int_{\Omega} \psi_p (v_n) (v_n - v_0) dx =\\ & =& \limsup_{n \to \infty} \int_{\Omega} (\psi_p (v_n) - \psi_p (v_0)) (v_n - v_0) dx \geq\\ &\geq& \limsup_{n \to \infty} \left[ \left(\int_{\Omega} |v_n|^p dx \right )^{1/p'}- \left( \int_{\Omega} |v_0|^p dx \right )^{1/p'} \right]\times\\ & &\times \left[ \left(\int_{\Omega} |v_n|^p dx \right)^{1/p} - \left( \int_{\Omega} |v_0|^p dx \right )^{1/p}\right] \geq 0 \end{eqnarray*} Hence $\|v_n\|_{X} \to \|v_0\|_{X}$, which together with (\ref{4.1}) yields the desired strong convergence. \quad$\Box$\medskip Let the function $g : \Omega \times \mathbb{R}^2 \to \mathbb{R}$ be a Carath\ eodory function, i.e. $g(x, \cdot, \cdot)$ is continuous for a.e. $x \in \Omega$ and $g(\cdot, \lambda, s)$ is measurable for all $(\lambda, s) \in \mathbb{R}^2$. Moreover, let $g(x, \lambda, 0)= 0$ for any $(x, \lambda) \in \Omega \times \mathbb{R}$ and given any bounded interval $J \subset \mathbb{R}$ we assume that there exists an exponent $q\in (p,p^{**})$ with $p^{**}=\frac{Np}{N-p}$ (for $N >2p$); $p^{**}=\infty$ (for $N \leq 2p$) such that for any $\varepsilon > 0$, there exists a constant $C_{\varepsilon}$ such that $$\label{4.3} |g(x,\lambda, s)| \leq \varepsilon |s|^{p-1}+ C_{\varepsilon} |s|^{q-1} \mbox{ for a.e. } x \in \Omega \mbox{ and all } \lambda \in J, s \in \mathbb{R}.$$ Note that (\ref{BPNp}) can be written in the equivalent form $$\label{4.4} \psi_p (v) = \lambda \Lambda \psi_p (\Lambda v) + \Lambda g (x, \lambda, \Lambda v).$$ Due to (\ref{4.3}) the right hand side of (\ref{4.4}) defines an operator $$T_{\lambda, g} : v \mapsto \lambda \Lambda \psi_p (\Lambda v) + \Lambda g (x, \lambda, \Lambda v)$$ from $X$ into $X^*$ which is compact. Indeed, by Lemma \ref{lem2.1} (i) we get $\Lambda v \in W^{2,p} (\Omega)$ and $\Lambda \psi_p (\Lambda v) \in W^{2,p'} (\Omega)$. Furthermore by using (\ref{4.3}) and the fact that $W^{2,p}(\Omega) \subset L^{q} (\Omega)$, we find that $\Lambda g (x, \lambda, \Lambda v) \in W^{2,q'} (\Omega)$. Thus $T_{\lambda, g}$ maps any bounded set of $X$ onto a bounded set of $W^{2,q'} (\Omega)$, which is compactly embedded in $X^*$, since $q < p^{**}$. Then this fact and Lemma \ref{lem4.1} imply that $\Psi_p - T_{\lambda, g}$ satisfies condition $(S_{+})$. So, given an open and bounded set $D \subset X$ such that $\Psi_p (v) - T_{\lambda, g} (v) \neq 0$ for any $v \in \partial D$, the generalized degree of Browder and Petryshin $$\Deg [\Psi_p - T_{\lambda, g}; D,0]$$ is well defined. \begin{lemma}\label{lem4.2} $\|\Lambda g (x, \lambda, \Lambda v)\|_{X^*} = o (\|v\|_X^{p-1})\mbox{ as } \|v\|_{X} \to 0$. \end{lemma} \paragraph{Proof.} Since $\Lambda$ is symmetric, we have $$\label{4.5} \|\Lambda g(x, \lambda, \Lambda v)\|_{X^*} = \sup\limits_{\|\varphi\|_{X} \leq 1} \int_{\Omega} \Lambda g (x, \lambda, \Lambda v) \varphi dx = \sup\limits_{\|\varphi\|_{X} \leq 1} \int_{\Omega} g (x, \lambda, \Lambda v) \Lambda \varphi dx.$$ Then, for any $\varepsilon > 0$, by virtue of (\ref{4.3}) and Lemma \ref{lem2.1} (i), we find \label{4.6} \begin{aligned} \Big| \int_{\Omega} g (x, \lambda, \Lambda v) \Lambda \varphi d x \Big| \leq & \int_{\Omega} \varepsilon |\Lambda v|^{p-1} |\Lambda \varphi| dx + \int_{\Omega} C_{\varepsilon} |\Lambda v|^{q-1} |\Lambda \varphi| dx\\ \leq& \varepsilon \|\Lambda v\|^{p-1}_{L^p} \|\Lambda \varphi\|_{L^p} + C_{\varepsilon} \|\Lambda v\|^{q-1}_{L^q} \|\Lambda \varphi\|_{L^q}\\ \leq &\varepsilon c^p_p \|v\|^{p-1}_{X} \|\varphi\|_{X} + C_{\varepsilon} c^q \|\Lambda v\|^{q-1}_{W^{2,p}} \|\Lambda \varphi\|_{W^{2,p}}\\ \leq& \varepsilon c^p_p \|v\|^{p-1}_{X}+ C_{\varepsilon} c^q c_p^q \|v\|_{X}^{q-1}, \end{aligned} where $c_p$ is the constant appearing in Lemma \ref{lem2.1} (i) and $c > 0$ is the embedding constant for $W^{2,p} (\Omega) \hookrightarrow L^q (\Omega)$. Thus the assertion follows from (\ref{4.5}) and (\ref{4.6}), since $p0$ be as in Lemma \ref{lem3.4} and consider $\lambda < \lambda_1 (p) + \delta, \lambda \neq \lambda_1 (p)$. Then Lemma \ref{lem4.2} and simple homotopy argument yields $$\label{4.7} \Deg [\Psi_p - T_{\lambda,g}; B_r (0), 0] = \Deg [\Psi_p - T_{\lambda,0} ; B_{\lambda} (0),0]$$ if $r > 0$ is chosen sufficiently small (cf. \cite{DKN}, \cite{D}, \cite{DEM}, \cite{DM} or [R]). Here $B_{r} (0)$ is the ball centred at the origin and with radius $r > 0$. \begin{lemma}\label{lem4.3} $\Deg [\Psi_p - T_{\lambda,0} ; B_{r} (0), 0] = \pm 1$ for $\lambda < \lambda_1 (p) + \delta, \lambda \neq \lambda_1 (p)$ and $sgn (\lambda_1 (p)- \lambda) = \pm 1$. \end{lemma} \paragraph{Proof.} To prove the jump'' of the degree we adopt the method developed in \cite{D} (see also \cite{DKN}). Thus we just sketch the proof and refer to [DKN, Theorem 3.7] or [D, Theorem 14.18] for the details. Consider the functional $$F_{\lambda} (v) = \frac{1}{p} \int_{\Omega} |v|^p dx - \frac{\lambda}{p} \int_{\Omega} |\Lambda v|^p dx.$$ It follows from the variational characterization of $\lambda_1 (p)$ (see Section 3) that for $\lambda < \lambda_1 (p)$ we have $$\langle F'_{\lambda} (v), v\rangle_X > 0$$ for $v \in \partial B_{r} (0)$ and $v= 0$ is the only critical point of $F_{\lambda}$ (here $\langle \cdot, \cdot \rangle_X$ denotes the duality between $X^{*}$ and $X)$ and hence $$\label{4.8} \Deg [\Psi_p - T_{\lambda,0}; B_r (0), 0] = 1$$ by the properties of the degree (cf.\cite{S}). Let now $\lambda \in (\lambda_1 (p), \lambda_1 (p) + \delta)$. As in (DKN, Theorem 3.7] we define a function $\eta : \mathbb{R} \to \mathbb{R}$ by $$\eta (t) = \left\{ \begin{array}{ll} 0, &\mbox{for } t < K, \\ \frac{2 \delta}{\lambda_{1}(p)} (t - 2 K), &\mbox{for }t \geq 3K, \end{array}\right.$$ The function $\eta (t)$ is continuously differentiable, positive and strictly convex in $(K, 3K), K > 0$. Let us modify $F_{\lambda}$ as follows $$\tilde{F}_{\lambda} (v): = F_{\lambda} (v) + \eta (\frac{1}{p} \int_{\Omega} |v|^p dx).$$ The properties of $\lambda_1 (p)$ stated in Theorem \ref{th1.1} now imply the following properties of $\tilde{F}_{\lambda} :$ \begin{itemize} \item $\tilde{F}_{\lambda}$ is continuously Fr\' echet differentiable and its critical point $v_0 \in X$ corresponds to a solution of the equation $$\psi_p (v_0) - \frac{\lambda}{1 + \eta' (\frac{1}{p} \int_{\Omega} |v_0|^p dx)} \Lambda \psi_p (\Lambda v_0) = 0.$$ \item For $\lambda \in (\lambda_1 (p), \lambda_1 (p) + \delta)$ the only nontrivial critical points of $\tilde{F}_{\lambda}$ occur if $$\eta' \left ( \frac{1}{p} \int_{\Omega} |v_0|^p dx \right ) = \frac{\lambda}{\lambda_1 (p)} - 1.$$ \item Due to the definition of $\eta$ we then have $$\frac{1}{p} \int_{\Omega} |v_0|^p dx \in (K, 3K)$$ and due to the simplicity of $\lambda_1 (p)$, either $v_0 = -t v_1 (p)$ or $v_0 = t v_1 (p)$, for some $t \in ((pK)^{1/p}, (3pK)^{1/p}), v_1 (p)$ as in the Section 3. \item $\tilde{F}_{\lambda}$ has precisely three isolated critical points $- t v_1(p), 0, t v_1 (p)$. \item $\tilde{F}_{\lambda}$ is weakly lower semicontinuous and even. \item $\tilde{F}_{\lambda}$ is coercive, i.e. $$\lim_{\|v\|_{X} \to \infty} \tilde{F}_{\lambda} (v) = \infty$$ \item $-t v_1 (p), t v_1 (p)$ are the points of the global minimum of $\tilde{F}_{\lambda}; 0$ is an isolated critical point of `saddle type''. \item $\langle \tilde{F}'_{\lambda} (v), v \rangle_X > 0$ for $\|v\|_X = R$ if $R > 0$ is large enough. \end{itemize} The properties of the degree now imply that for small $\rho > 0$ and large $R >0$ we have $$\Deg [\tilde{F}'_{\lambda}; B_{\rho} (\pm t v_1(p)),0] = \Deg [\tilde{F}'_{\lambda} ; B_R (0), 0]=1\,.$$ The additivity property of the degree then yields for $0< r< (pK)^{1/p}$, $$\label{4.9} \Deg [\Psi_p - T_{\lambda,0}; B_r (0), 0] = \Deg [\tilde{F}'_{\lambda}; B_r (0), 0] = -1\,.$$ The assertion of Lemma \ref{lem4.3} follows now from (\ref{4.8}) and (\ref{4.9}). \quad$\Box$\medskip If we combine (\ref{4.7}) with Lemma \ref{4.3} we come to the following conclusion: for $r>0$ sufficiently small $$\Deg [\Psi_p - T_{\lambda, g} ; B_r (0), 0] = \pm 1$$ for $\sgn (\lambda_1 (p) - \lambda) = \pm 1$. Following the proof of [R, Theorem \ref{th1.3}] we prove that continuum of nontrivial solutions $(\lambda, v) \in \mathbb{R} \times X$ of (\ref{4.4}) bifurcates from $(\lambda_1 (p), 0)$ and it is either unbounded in $\mathbb{R} \times X$ or meets the point $(\lambda_e (p), 0)$, where $\lambda_e (p) > \lambda_1 (p)$ is an eigenvalue of (\ref{3.1}). The assertion of Theorem \ref{th1.2} now follows from the fact that $(\lambda, u)$ solves $\rm{(BP_N)_p}$ if and only if $(\lambda, -\Delta u)$ solves (\ref{4.4}). \section{One-dimensional problem} Let $N = 1$ and $\Omega = (0,1)$. Then $\rm{(E_N)_p}$ reduces to (\ref{E1p}) and obviously the assertions of Theorems \ref{th1.1}, \ref{th1.2} remain true. We point out that $W^{2,p}(0,1) \hookrightarrow \hookrightarrow C^1([0,1])$ in the case $N = 1$, and so $\psi_p(v) \in C^1 ([0,1])$, $v (0) = v (1) = 0$ for any solution $v$ of (\ref{2.3}). Hence we do not need Lemmas 3.1 and 3.3 in this case. For the sake of brevity we shall write $\lambda_1 : = \lambda_1 (p), u_1 : = u_1 (p)$. It follows from the symmetry of (\ref{E1p}) and Theorem \ref{th1.1} (simplicity of $\lambda_1$) that $u_1 (t) = u_1 (1-t)$ for $t \in [0,1]$, i.e. $u_1$ is even with respect to $\frac{1}{2}$. Making use of this observation, we give a precise description of all eigenvalues and eigenfunctions of ${\rm(E_1)_p}$. Indeed, set $$\begin{gathered} u_n (t) = u_1 (n t);t \in [0, \frac{1}{n}],\\ u_n (t) = -u_1 (nt -1), t \in [\frac{1}{n}, \frac{2}{n}],\\ \dots\\ u_n(t) = (-1)^n u_1 (nt - n + 1), t \in [\frac{n-1}{n}, 1]. \end{gathered}$$ Then $u_n = u_n (t), t \in [0,1]$, is an eigenfunction of (\ref{E1p}) associated with the eigenvalue $\lambda_n = n^{2p} \lambda_1$. On the other hand, let $u= u (t)$ be an eigenfunction of ${\rm(E_1)_p}$ associated with some eigenvalue $\lambda_e$. According to Theorem \ref{th1.1} it must be $\lambda_e > \lambda_1$ and $u$ changes sign in $(0,1)$. By Lemma A.4 the number of nodes of $u$ in (0,1) is finite. Assume first that $\lambda_e =\lambda_n$, for some $n>1$. Let us normalize $u$ as follows: $u'(0) = u'_n (0)>0$. Note that since $u$ and $u_n$ are oscillatory, we must have, according to Lemma A.3, that $$(\psi_p (u''(t)))'|_{t=0} < 0 \quad\mbox{and}\quad (\psi_p (u''_n (t)))'|_{t=0} < 0,$$ respectively. Let $(\psi_p (u''(t)))'|_{t=0} = (\psi_p (u_n'' (t)))'|_{t=0}$. Then Lemma A.1 implies that $u(t) = u_n (t), t \in [0,1]$. Let $(\psi_p (u''(t)))'|_{t=0} \neq (\psi_p (u_n''(t)))'|_{t=0}$. Then Lemma A.2 implies that $u(1) \neq 0$, a contradiction. Let $\lambda_e \neq \lambda_k$ for any $k \geq 2$. Define $$\begin{gathered} \tilde{u} (t) = u_1 \left( \left(\frac{\lambda_e}{\lambda_1}\right) ^{1/(2p)} t \right), t \in \left[0, \left(\frac{\lambda_1}{\lambda_e}\right)^{1/(2p)} \right], \\ \tilde{u} (t) = - u_1 \left( \left(\frac{\lambda_e}{\lambda_1}\right)^{1/(2p)} t - 1 \right), t \in \left[ \left(\frac{\lambda_1}{\lambda_e}\right)^{1/(2p)}, 2 \left( \frac{\lambda_1}{\lambda_e}\right)^{1/(2p)} \right], \mbox{ etc.}\end{gathered}$$ Then $\tilde{u} (1) \tilde{u}''(1) < 0$. Let us normalize $u$ as $u' (0) = \tilde{u}' (0) > 0$. Then it follows from Lemma A.2 that $u(1) = u''(1) = 0$ cannot hold at the same time. Thus Theorem \ref{th1.3} is proved. Let $X = C([0,1])$. Let $g : [0,1] \times \mathbb{R}^2 \to \mathbb{R}$ be a continuous function satisfying $g(t, \lambda, 0) = 0$ for any $(t, \lambda) \in (0,1) \times \mathbb{R}$ and given any bounded interval $J \subset \mathbb{R}$ we assume that $$\label{5.1} |g(t, \lambda , s)| = o (|s|^{p-1})$$ holds near $s = 0$ uniformly for all $(t, \lambda) \in [0,1] \times J$. Note that ${\rm(BP_1)_p}$ can be written in the equivalent form $$\label{5.2} v = \psi_{p'} (\lambda \Lambda \psi_p (\Lambda v) + \Lambda g (t, \lambda, \Lambda v)).$$ Due to Lemma \ref{lem2.1} (i), the right hand side of (\ref{5.2}) defines an operator $$R_{p,\lambda, g} : (p, \lambda, v) \mapsto \psi_{p'} (\lambda \Lambda \psi_p (\Lambda v) + \Lambda g (t, \lambda, \lambda v))$$ which is compact from $(1, \infty) \times \mathbb{R} \times X$ into $X$. If $I : X \to X$ denotes the identity mapping, the Leray-Schauder degree $$\deg [I - R_{p, \lambda, g} ; D, 0]$$ is well defined for any open bounded set $D$ such that $v - R_{p,\lambda, g} (v) \neq 0$ for $v \in \partial D$. \begin{lemma}\label{lem5.1} Let $\lambda \neq \lambda_n$. Then there is $r > 0$ (sufficiently small) such that $$\label{5.3} \deg [I - R_{p, \lambda, g} ; B_r (0), 0] = \deg [I - R_{p,\lambda ,0}; B_r (0) , 0].$$ \end{lemma} \paragraph{Proof.} Standard argument based on (\ref{5.1}) yields that the homotopy $$H(\tau,v)= v - \psi_{p'} (\lambda \Lambda \psi_p (\Lambda v) + \tau \Lambda g (t, \lambda, \Lambda v))$$ satisfies $H (\tau, v) \neq 0$ for all $\tau \in [0,1]$ and $v \in \partial B_r (0)$ if $r> 0$ is small enough. So (\ref{5.3}) follows from the homotopy invariance property of the Leray-Schauder degree. \quad$\Box$ \medskip Let $\lambda \in (\lambda_n (p), \lambda_{n+1} (p)), n = 0, 1, 2, \ldots$, where we set $\lambda_0 (p) = - \infty$ and $\lambda_1 (p), \lambda_2 (p), \ldots$ are as above, then we have. \begin{lemma}\label{lem5.2} $\deg [I - R_{p,\lambda, 0} ; B_r (0), 0] = (-1)^n$. \end{lemma} \paragraph{Proof.} We follow the idea in \cite{DEM}. Note that it follows from Theorems 1.1, 1.3 that $$\lambda_n : p \mapsto \lambda_n (p), n = 1, 2, \ldots ,$$ are continuous functions on $(1, \infty)$. Assume that $p < 2$. Define $\lambda(q), q \in [p, 2]$, by the following way \begin{aligned} \lambda (q) : =& \frac{\lambda - \lambda_n (p)}{\lambda_{n+1} (p) - \lambda_n (p)} \cdot (\lambda_{n+ 1} (q) - \lambda_n (q)) + \lambda_n (q),\quad n \geq 1,\\ \lambda(q) : =& \lambda_1 (q) - (\lambda_1 (p) - \lambda), n = 0. \end{aligned} Then $$H (q, v) : = v - R_{q, \lambda (q), 0} (v)= v - \psi_{q'} (\lambda (q) \Lambda \psi_q (\Lambda v))$$ satisfies $H (q,v) \neq 0$ for all $q \in [p,2]$ and $v \in \partial B_r (0)$. It follows from the homotopy invariance property of the Leray-Schauder degree that $$\label{5.4} \deg [I - R_{p, \lambda, 0}; B_r (0), 0] = \deg [ I - R_{2, \lambda (2), 0}; B_r (0),0].$$ The same approach but in the interval $[2,p]$ yields to the same conclusion also for $p>2$. Since $\lambda_n (2) < \lambda (2) < \lambda_{n+ 1} (2)$, the classical Leray-Schauder index formula implies that $$\label{5.5} \deg [I - R_{2, \lambda (2), 0}; B_r (0) , 0] = (-1)^n.$$ The assertion of lemma follows now from (\ref{5.4}) and (\ref{5.5}). \quad$\Box$\medskip With Lemmas \ref{lem5.1} and \ref{lem5.2} in hand we can follow the proof of [R, Theorem \ref{th1.3}] to prove that continua of nontrivial solutions $(\lambda, v) \in \mathbb{R} \times X$ of (\ref{5.2}) bifurcate from $(\lambda_n (p), 0), n = 1,2 \ldots$, and they are either unbounded in $\mathbb{R} \times X$ or meet the point $(\lambda_{m}(p), 0)$ with $m \neq n$. The assertion of Theorem \ref{th1.4} follows from the fact that $(\lambda, u)$ solves (\ref{BP1p}) if and only if $(\lambda, - \Delta u)$ solves (\ref{5.2}). \section{Appendix} To justify some statements in Section 5 we present here a brief study of the initial value problem associated with the equation in $(E_1)_p$ with $\lambda > 0$: $$\begin{gathered} u'' = \psi_{p'} (w), \quad u (t_0) = \alpha, \quad u' (t_0) = \beta, \\ w'' = \lambda \psi_p (u), \quad w (t_0)= \gamma, \quad w' (t_0) = \delta\,. \end{gathered} \label{A1}$$ By a {\it solution} of (\ref{A1}) we understand a couple of functions $(u,w)$ which are of class $C^2$ and fulfil the equations and initial conditions in (\ref{A1}). \begin{lemma} \label{lemA1} The solution to (\ref{A1}) is locally unique. \end{lemma} \paragraph{Proof.} Without loss of generality we can restrict ourselves to $t_0 = 0$ and $p\in (1,2)$ (the case $p>2$ is treated similarly). Local existence is a consequence of the Schauder fixed point theorem. For its uniqueness we have to distinguish among several cases: \begin{itemize} \item[(I)] $\alpha \neq 0$ implies that both functions $\psi_p (u(t))$ and $\psi_{p'}(w(t))$ are of class $C^1$ in the neighbourhood of $t = 0$ and so the assertion follows from the classical theory. \item[(II)] $\alpha = 0$, in this case $\psi_p (u(t))$ is not $C^1$ in $t= 0$. \item[(II)(i)] $\alpha = 0, \beta \neq 0$. Let $(u, w_1), (v , w_2)$ be two solutions of (\ref{A1}) in $(0, \varepsilon)$ with some $\varepsilon > 0$. Then $$\psi_p (u'' (t)) - \psi_p (v'' (t)) = \lambda \int^{t}_{0} (t - \tau) (\psi_p (u(\tau)) - \psi_p (v (\tau)))d\tau. \label{A2}$$ By the assumption, $\frac{u (\tau)}{\tau}, \frac{v(\tau)}{\tau}$ lie in the neighbourhood of $\beta \neq 0$ for $\tau \in (0, \varepsilon)$ wiht $\varepsilon$ small enough. We thus have $K_1 > 0$ such that $$\left| \psi_p \left( \frac{u (\tau)}{\tau}\right) - \psi_p \left(\frac{v (\tau)}{\tau}\right)\right| \leq K_1 \left|\frac{u(\tau)}{\tau} - \frac{v (\tau)}{\tau}\right|, \label{A3}$$ $\tau \in (0, \varepsilon), K_1$ independent of $\varepsilon <<1$. On the other hand there is $K_2 > 0$ such that $$\left| \psi_p (u''(t)) - \psi_p (v'' (t))\right| \geq K_2 |u''(t) - v'' (t)|, \label{A4}$$ $t \in (0, \varepsilon)$. Now, it follows from (\ref{A2})--(\ref{A4}) $$K_2 |u'' (t) - v'' (t)| \leq \lambda \int^{t}_0 (t - \tau)\tau^{p-1} K_1 \left|\frac{u (\tau)}{\tau} - \frac{v (\tau)}{\tau}\right| d \tau.$$ Taking into account $$u (\tau) - v (\tau) = \int^{\tau}_{0} (\tau- \sigma) (u'' (\sigma) - v'' (\sigma) ) d \sigma,$$ we arrive at $$\|u'' -v''\|_{\varepsilon} \leq \lambda \frac{K_1}{K_2} \varepsilon^{p+2} \|u'' - v''\|_{\varepsilon}, \label{A5}$$ where $\| \cdot \|_{\varepsilon}$ is the $\sup$ norm on $[0, \varepsilon]$. This implies $u = v$ (and thus $w_1 = w_2$) for $\varepsilon$ small enough. \item[(II) (ii)] $\alpha = \beta = 0, \gamma \neq 0$ and (iii) $\alpha = \beta = \gamma = 0, \delta \neq 0$. Instead of (\ref{A2}) we use the following fact $$\psi_{p'} (w_1''(t)) - \psi_{p'}(w_2''(t)) = \psi_{p'} (\lambda) \int^t_0 (t - \tau) (\psi_{p'} (w_1 (\tau)) - \psi_{p'} (w_2 (\tau))) d \tau. \label{A2'}$$ Since $p' > 2$, we have $$|\psi_{p'} (w_1 (\tau)) - \psi_{p'}(w_2 (\tau)) |\leq K_1 |w_1 (\tau) - w_2 (\tau)|,$$ $\tau \in (0, \varepsilon)$. Hence $$\Big| \int^t_0 (t - \tau) (\psi_{p'} (w_1 (\tau)) - \psi_{p'} (w_2 (\tau))) d \tau \Big| \leq K_1 \varepsilon^2 \|w_1 - w_2\|_{\varepsilon}. \label{A3'}$$ It follows from the initial conditions that $\frac{w''_i(t)}{t^{2(p-1)}}$, $i =1, 2$, lie near \break $\lambda \gamma \psi_p (\frac{1}{2}) \neq 0$ in the case (ii) and $\frac{w_i (t)}{t^{2p - 1}}, i = 1,2$, lie near $\lambda \delta \psi_p\left (\frac{1}{p'(p'+1)}\right) \neq 0$ in the case (iii). Hence there exists $K_2 > 0$ such that $$\left| \psi_{p'}\left ( \frac{w_1'' (t)}{t^{2(p-1)}}\right) - \psi_{p'} \left( \frac{w_2'' (t)}{t^{2(p-1)}} \right)\right| \geq K_2 \left| \frac{w_1''(t)}{t^{2(p-1)}} - \frac{w_2'' (t)}{t^{2(p-1)}}\right| \label{A4'}$$ in the case (ii) and $$\left| \psi_{p'}\left ( \frac{w_1'' (t)}{t^{2p-1}}\right) - \psi_{p'} \left( \frac{w_2'' (t)}{t^{2p-1}} \right)\right| \geq K_2 \left| \frac{w_1''(t)}{t^{2 p-1}} - \frac{w_2'' (t)}{t^{2 p-1}}\right| \label{A4''}$$ in the case (iii). Taking into account $$w_1 (t) - w_2 (t) = \int^t_0 (t - \tau) (w_1'' (\tau) - w_2'' (\tau)) d \tau$$ we derive from (\ref{A2'}), (\ref{A3'}), (\ref{A4'}) and (\ref{A4''}) that $$\|w_1 - w_2\|_{\varepsilon} \leq \frac{K_1}{K_2} \psi_{p'}(\lambda) \varepsilon^{2p + 2} \|w_1 - w_2\|_{\varepsilon}$$ in the case (ii) and $$\|w_1 - w_2\|_{\varepsilon} \leq \frac{K_1}{K_2} \psi_{p'}(\lambda) \varepsilon^{2p + 3} \|w_1 - w_2\|_{\varepsilon}$$ in the case (iii). \item[(II)(iv)] $\alpha = \beta = \gamma = \delta = 0$. In this case (\ref{A1}) has always the trivial solution $u_0 = w_0 = 0$. Let $(u, w)$ be a nontrivial solution. Then $$|\psi_p (u'' (t))|\leq \lambda \int^t_0 (t - \tau) \psi_p (|u (\tau)|) d \tau \leq \lambda \varepsilon^2 \|u\|^{p-1}_{\varepsilon}, t \in (0, \varepsilon),$$ which yields $$\|u''\|_{\varepsilon}^{p-1} \leq \lambda \varepsilon^{2p} \|u''\|_{\varepsilon}^{p-1} ,$$ i.e. $u = w = 0$. This completes the proof. \quad$\Box$ \end{itemize} \begin{lemma}\label{lemA2} Let $(u,w)$ and $(\tilde{u}, \tilde{w})$ be solutions of (\ref{A1}) defined on [0,1], respectively, $u(0) = w (0) = \tilde{u} (0) = \tilde{w} (0) = 0$, $u'(0) =\tilde{u}' (0) >0$, $w' (0) < \tilde{w}' (0)$. Then $u (t) < \tilde{u} (t)$ and $w(t) < \tilde{w} (t)$ for any $t \in (0,1]$. \end{lemma} \paragraph{Proof.} Assume that the assertion is not true . Then it follows from Lemma A.1 that there is $t_1>0$ such that $u(t_1) = \tilde{u} (t_1)$ and $u (t) < \tilde{u} (t), t \in (0,t_1)$. Simultaneously, the fact that both $u$ and $\tilde u$ solve $(E_1)_p$ imply that \begin{eqnarray*} \lefteqn{\int^{t_1}_{0} (t_1- \tau) \psi_{p'} \Big( \lambda \int^{\tau}_{0} (\tau - \sigma)\psi_p (u (\sigma))d \sigma + w' (0) \tau\Big) d \tau }\\ &=& \int^{t_1}_0 (t_1 - \tau) \psi_{p'} \Big(\lambda \int^{\tau}_0 (\tau - \sigma)\psi_p (\tilde{u} (\sigma)) d \sigma + \tilde{w}' (0) \tau \Big)d \tau \end{eqnarray*} which contradicts the monotone character of the functions $\psi_p$ and $\psi_{p'}$. The same argument applies for $w$ and $\tilde{w}.$ \quad$\Box$ \begin{lemma} \label{lemA3} Let $(u,w)$ be a nonzero solution of (\ref{A1}) defined on $[0,1]$ and satisfying $u(0) = w(0) = u(1) =w(1)=0$. Then $u'(0)w'(0)<0$. \end{lemma} \paragraph{Proof.} Multiply the first (second) equation in (\ref{A1}) by $w' (u')$ and add to get $$u'(x)w'(x) = \frac{|w(x)|^{p'}}{p'} + \lambda \frac{|u(x)|^p}{p} - C \quad\mbox{for all}\quad x\in [0,1]. \label{A6}$$ Let $x_0 \in (0,1)$ be the point satisfying $$|u(x_0)|=\max_{x\in[0,1]} |u(x)| >0\,.$$ Then (\ref{A6}) implies $$0=\frac{|w(x_0)|^{p'}}{p'} + \lambda \frac{|u(x_0)|^p}{p} - C,$$ i.e. $C>0$. Hence $u'(0)w'(0) < 0$ by (\ref{A6}). \quad$\Box$ \begin{lemma} \label{lemA4} Let us assume the same as in the previous lemma. Then $u$ (and also $w$) changes sign in $(0,1)$ at most finitely many times. \end{lemma} \paragraph{Proof.} Let $u$ have an infinite number of bumps in $(0,1)$. Then there exist sequences $x_n, y_n$ such that $u(x_n) =u'(y_n) = 0, x_n \rightarrow x_0, y_n \rightarrow x_0, x_n, y_n, x_0 \in [0,1]$. Then $u(x_0)= u'(x_0) =0$, hence (\ref{A6}) gives $$0=\frac{|w(x_0)|^{p'}}{p'} - C.$$ Since $C>0$, we have $$w(x_0)>0 \quad \mbox{or}\quad w(x_0) < 0.$$ Due to $$u'(x) = \int^x_{x_0} \psi_{p'} (w(y)) dy + \psi_{p'} (w(x_0)),$$ the function $u'(x)$ should be of definite sign in a neighbourhood of $x=x_0$, which contradicts the observation that $u'(y_n) = 0, y_n \rightarrow x_0$. \quad$\Box$ \paragraph{Acknowledgements} The first author is partially supported by grant number 201/00/0376 from the Grant Agency of the Czech Republic. The second author is supported by grant number 09440070 from the Grant-in-Aid for Scientific Research, Ministry of Education, Science, Sports and Culture, Japan and by Waseda University Grant number 99B-013 for Special Research Projects. \begin{thebibliography}{0} \frenchspacing \bibitem{A} A. Anane: {\it Simplicit\' e et isolation de la premi\' ere valeur propre du p-Laplacien avec poids,} C. R. Acad. Sci. 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Trudinger: {\it Elliptic Partial Differential Equations of Second Order}, Second Ed., Springer-Verlag, Berlin Heidelberg New York Tokyo 1983. \bibitem{IO} T. Idogawa, M. \^Otani: {\it The first eigenvalues of some abstract elliptic operator}, Funkcialaj Ekvacioj 38 (1995), 1--9. \bibitem{R} P. H. Rabinowitz: {\it Some global results for nonlinear eigenvalue problems,} J. Funct. Anal. 7 (1971), 487--513. \bibitem{S} I. V. Skrypnik:{\it Nonlinear Elliptic Boundary Value Problems, Teubner,} Leipzig 1986. \end{thebibliography} \noindent\textsc{Pavel Dr\'abek}\\ Centre of Applied Mathematics \\ University of West Bohemia \\ Univerzitn\'\i 22, 306 14 Plze\v n\\ Czech Republic\\ e-mail: pdrabek@kma.zcu.cz \smallskip \noindent\textsc{Mitsuharu \^Otani}\\ Department of Applied Physics \\ School of Science and Engineering \\ Waseda University \\ 3-4-1, Okubo Tokyo, Japan, 169-8555 \\ e-mail: otani@mn.waseda.ac.jp \end{document}