%this is a latex file \documentstyle[twoside]{article} \pagestyle{myheadings} \markboth{\hfil THE LAZER McKENNA CONJECTURE \hfil EJDE--1993/07}% {EJDE--1993/07\hfil Alfonso Castro and Sudhasree Gadam \hfil} \begin{document} \ifx\Box\undefined \newcommand{\Box}{\diamondsuit}\fi \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}\newline Vol. 1993(1993), No. 07, pp. 1-6. Published October 30, 1993.\newline ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113} \vspace{\bigskipamount} \\ The Lazer Mckenna Conjecture for Radial Solutions in the $R^N$ Ball \thanks{ {\em 1991 Mathematics Subject Classifications:} Primary 34B15, Secondary 35J65.\newline\indent {\em Key words and phrases:} Lazer-McKenna conjecture, radial solutions, jumping \newline\indent nonlinearities. \newline\indent \copyright 1993 Southwest Texas State University and University of North Texas\newline\indent Submitted: May 2, 1993.\newline\indent Partially supported by NSF grant DMS-9246380. } } \date{} \author{Alfonso Castro and Sudhasree Gadam} \maketitle \begin{abstract} When the range of the derivative of the nonlinearity contains the first $k$ eigenvalues of the linear part and a certain parameter is large, we establish the existence of 2k radial solutions to a semilinear boundary value problem. This proves the Lazer McKenna conjecture for radial solutions. Our results supplement those in [5], where the existence of $k + 1$ solutions was proven. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{remark}{Remark}[section] \newtheorem{corollary}{Corollary}[section] \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \section{Introduction} Here we consider the boundary value problem \begin{eqnarray} &-\Delta u(x) = g(u(x)) + t\varphi (x) + q(x)\mbox{ for } x\in\Omega& \label{1.1}\\ &u(x) = 0 \mbox{ for } x\in \partial \Omega\,, & \label{1.2}\end{eqnarray} where $\Delta$ denotes the Laplacean operator, $\Omega$ is a smooth bounded region in $R^N (N>1)$, $g$ is a differentiable function, $q$ is a continuous function, and $\varphi >0$ on $\Omega$ is an eigenfunction corresponding to the smallest eigenvalue of $-\Delta$ with zero Dirichlet boundary condition. We will assume that $$\lim_{u\rightarrow-\infty}\frac{g(u)}{u} = \alpha\quad \mbox{ and }\quad \lim_{u\rightarrow\infty }\frac{g(u)}{u} = \beta\,. \label{1.3}$$ Motivated by the classical result of A. Ambrosetti and G. Prodi [1], equations of the form (\ref{1.1})--(\ref{1.2}) have received a great deal of attention when the interval $(\alpha, \beta)$ contains one or more eigenvalues of $-\Delta$ with zero Dirichlet boundary data. In [1] it was shown that when $(\alpha, \beta)$ contains only the smallest eigenvalue then for $t < 0$ large enough the equation (\ref{1.1})--(\ref{1.2}) has two solutions. Upon considerable research on extensions of this result, A. C. Lazer and P. J. McKenna conjectured that when $(\alpha, \beta)$ contains the first $k$ eigenvalues then (1.1)--(1.2) has $2k$ solutions. Here we prove that such a conjecture is true if one restricts to radial solutions ($u(x) = u(y)$ if $||x|| = ||y||$) in a ball. This conjecture, however, is not true in general. In [7] E. N. Dancer gives an example where $(\alpha, \beta)$ contains more than two eigenvalues and yet (\ref{1.1})--(\ref{1.2}) has only four solutions for $t < 0$ large. The reader is referred to [13] for an extensive review on problems with jumping nonlinearities and their applications to the modeling of suspension bridges. Throughout this paper $[x]$ denotes the largest integer that is less than or equal to $x$. Our main result is stated as follows: \begin{theorem} Let $\Omega$ be the unit ball in $R^{N} (N > 1)$ centered at the origin. Let $0< \rho_1< \rho_2<\cdots <\rho_n< \cdots \rightarrow \infty$ denote the eigenvalues of $-\Delta$ acting on radial functions that satisfy (\ref{1.2}). If $$\alpha < \rho_1([j/2] + 1)^2 < \rho_k < \beta < \rho_{k+1}\label{1.5}\label{1.4}$$ and $q$ is radial function, then for $t$ negative and of sufficiently large magnitude, problem (\ref{1.1})--(\ref{1.2}) has at least $2(k - j)$ radial solutions, of which $k - j$ satisfy $u(0)>0$. \label{thm:1}\end{theorem} This theorem with $j = 1$ proves the Lazer-McKenna conjecture in the class of radial functions. Theorem \ref{thm:1} extends the results of D. Costa and D. de~Figueiredo (See [5]) since we do not require $\alpha < \rho _1$ and for any $N > 1$ we obtain $k$ solutions with $u(0) > 0$. In [5] the authors proved, only for $N = 3$, that the equation (\ref{1.1})--(\ref{1.2}) has $k$ solutions with $u(0) > 0$. The reader is also referred to [14] for a study on the case $t > 0$. For other results on problems with jumping nonlinearities see [8], [11], [13] and references therein. For the sake of simplicity we will assume that $\alpha> 0$. Minor modifications needed for the case $\alpha \leq 0$ are left to the reader. \section{Preliminaries} \setcounter{equation}{0} Since $\varphi$ is a radial function, using polar coordinates $(r=\|x\|,\theta)$ we see that finding radial solutions to (\ref{1.1})--(\ref{1.2}) is equivalent to solving the two point boundary value problem \begin{eqnarray} &u'' + (\frac{N-1}{r})u' + g(u(r)) + t\varphi (r) + q(r) = 0\quad r\in [0,1]\,,\label{2.1}\\ &u'(0) = 0\,,\label{2.2}\\ & u(1) = 0\,,&\label{2.3} \end{eqnarray} where the symbol $'$ denotes differentiation with respect to $r = \|x\|$, $\varphi(r) \equiv \varphi(x)$, and $q(r) \equiv q(x)$. Let $\tau (\varphi ,q) = \tau$ be such that if $t < \tau$ then the problem (\ref{1.1})--(\ref{1.2}) has a positive solution $U_t := U$ (See [5], [11]). Following the ideas in [14] we will seek solutions to (\ref{1.1})--(\ref{1.2}) of the form $U + w$. It is easily seen that $U + w$ satisfies (\ref{1.1})--(\ref{1.2}) if and only if $w$ satisfies \begin{eqnarray} &w'' + \frac{N - 1}{r}w' + \lambda [g(U(r) + w(r)) - g(U(r))] = 0\,, r\in [0,1]&\label{2.4}\\ &w'(0) = 0\,,&\label{2.5}\\ &w(1) = 0\,, &\label{2.6}\end{eqnarray} for $\lambda = 1$. We will denote by $w:=w(\cdot ,t,\lambda ,d)$ the solution to (\ref{2.4})--(\ref{2.5}) satisfying $w(0) = d$. We prove Theorem \ref{thm:1} by studying the bifurcation curves for the equations (\ref{2.4})--(\ref{2.6}). For future reference we note that, for fixed $t\in R$, the set $$S \subset \{(\lambda ,w) \in R \times (C(\Omega) - \{0\})\,;\ (\lambda ,w)\mbox{ satisfies (\ref{2.4})--(\ref{2.6})}\}$$ is connected if and only if $\{(\lambda ,w(0))\,;\ (\lambda ,w)\in S\}$ is connected. This is an immediate consequence of the continuous dependence on initial conditions of the solutions to (2.4). In order to facilitate the proofs of the above theorems, we identify $S$ with the latter subset of ${R}^2$. We consider solutions to (\ref{2.4})--(\ref{2.6}) bifurcating from the set $\{(\lambda ,0)$; $\lambda >0\}$, which clearly is a set of solutions. Since the eigenvalues of the problem \begin{eqnarray} &z''+\frac{N-1}{r}z'+\lambda g'(U)z = 0\, \quad r\in [0,1]&\label{2.7}\\ &z'(0) = 0\,,&\label{2.8}\\ &z(1) = 0\,,&\label{2.9}\end{eqnarray} are simple, by general bifurcation theory (See [5]) it follows that if $\mu$ is an eigenvalue of (\ref{2.7})--(\ref{2.9}) then near $(\mu ,0)$ there are solutions to (\ref{2.4})--(\ref{2.6}) of the form $(\mu+ o(s),s\psi+ o(s))$ where $\psi \neq 0$ is an eigenfunction corresponding to the eigenvalue $\mu$. Given $t$, hence $U$, we will denote by $\mu_1 <\mu_2<\cdots\rightarrow \infty$ the eigenvalues to (\ref{2.7})--(\ref{2.9}). Now we are ready to establish the estimates on the points of bifurcation of (\ref{2.4})--(\ref{2.6}). \begin{lemma} If $\lim_{u\rightarrow+\infty} g(u)/u = \gamma$ then for any positive integer $j$ and $\epsilon > 0$ there exists $T(j)$ such that if $t < T$ then $\mu _j < (\rho _j/\gamma - \epsilon )$ \end{lemma} \noindent {\bf Proof}. Since $U$ tends to $\infty$ uniformly on compact subsets of [0,1) as $t\rightarrow -\infty$, by the Courant-Weinstein minmax principle we have $$\mu_j \leq \sup_{u\in M-\{0\}} ({\int}_\Omega \ \nabla u\cdot \nabla u )/({\int}_\Omega g'(U) u^2 )\,, \label{2.10}$$ where $M$ is any $j$-dimensional linear subspace. On the other hand, letting $M$ be the span of $\{\varphi _1,$ $...,\varphi _j\}$, where $\varphi _i$ is an eigenfunction corresponding to the eigenvalue $\rho _i$ we see that the numerator in the the right hand side of (\ref{2.10}) is less than or equal to $\rho _j\int_\Omega u^2$. This implies that $\mu_j <(\rho _j/(\gamma-\epsilon))$ for $t \ll 0$, which proves the lemma. Let $E(r, t, \lambda, d) := E(r) = ((w'(r, t, \lambda, d))^2/2) + \lambda \cdot (G(r, t, w(r, t, \lambda, d)))$, where $G(r, t, s)$ $= \int_{0}^{s} (g(U(r) + x) - g(U(r)))\,dx$. Because of (\ref{1.3}), arguing as in [2] (See also [4]), we see that for each $t$ and $\lambda$ in bounded sets $$E(r,t,\lambda ,d)\rightarrow +\infty\mbox{ uniformly on [0,1] as |d| tends to infinity.}\label{2.11}$$ \begin{remark} By the uniqueness of solutions to the initial value problem (\ref{2.4})--(\ref{2.5}), $w(0) = d$, we see that if $w(s) = w'(s) = 0$ for some $s\in [0,1]$ then $w(r) = 0$ for all $r\in [0,1].$ \end{remark} \begin{lemma} Let $t < \tau$ be given with $\alpha$ as in Theorem~\ref{thm:1}. If $\{(\lambda _n,w_n)\}$ is a sequence of solutions to (\ref{2.4})--(\ref{2.6}) such that for each $n$ $w_n$ has exactly $j$ zeros in (0,1), $\{\lambda _n\}$ converges to $\Lambda$, and $\{|w_n(0)|\}$ converges to infinity, then $$\alpha\Lambda \geq ([j/2] + 1)^2 \rho _1\,.$$\label{lm:3}\end{lemma} \noindent {\bf Proof:} Without loss of generality we can assume that $w_n(0) > 0$ for all $n$. Let $0 < r_{1,n} < \cdots< r_{k,n} < 1$ denote the zeros of $w_n$ in (0,1]. For $i = 1,\cdots,k$, let $s_{i,n} \in (r_{i,n},r_{i+1,n})$ be such that $$|w_n(s_{i,n})| = \max\{|w_n(t)|; t \in [r_{i,n},r_{i+1,n}]\}\,.$$ Since $g$ is locally Lipschitzian, by the uniqueness of solutions to initial value problems we see that $|w_n(s_{i,n})| \not = 0$. Thus $w'_n(s_{i,n}) = 0$ By (\ref{2.11}) we see that $\{w_n(s_{i,n})\}$ converges $to -\infty$ as $n$ tends to infinity. Now we analyze $w_n$ on $[s_{i,n},r_{i+1,n})$, for $i$ odd. By the definition of $\alpha$ we see that $g(x) = \alpha x + h(x)$ with $\lim_{x\rightarrow -\infty} h(x)/x = 0$, for $x < 0$. Let $s$ denote a limit point of $\{s_{i,n}\}$ and $b$ a limit point of $\{r_{i.n}\}$. Thus $\{z_n := w_n/w_n(s_{i,n})\}$ converges, uniformly on $[s,b]$, to the solution to \begin{eqnarray} &z''+\frac{N-1}{r}z'+\Lambda \alpha z = 0\,,\ r\in [s,b]& \label{2.12}\\ &z(s) = 1\,,\ z'(s) = 0\,.\label{2.13}\end{eqnarray} By the Sturm Comparison Theorem we know that $z > 0$ on $[s,s + (\rho _1/(\Lambda\alpha ))]$. Hence for $\delta > 0$ sufficiently small there exists $\eta$ such that if $n > \eta$ then $w_n < 0$ on $[s_{i,n},s_{i,n} + (\rho _1/(\Lambda\alpha )) - \delta ]$. Since this argument is valid for all $i$ odd, we see that $$m(\{x;w_n(x)<0\})>([k/2]+1) ((\frac{\rho _1}{\Lambda\alpha})^{1/2} -\delta)\,,$$ which proves the lemma. \begin{corollary} Let $t < \tau$. If $\{(\lambda _n,w_n)\}$ is a sequence of solutions to (\ref{2.4})--(\ref{2.6}), $w_n$ has exactly $k$ zeros in (0,1) for each $n$, $\{\lambda _n\}$ converges to $\Lambda$, and $\{|w_n(0)|\}$ converges to infinity, then $(\alpha+\beta )\Lambda \geq ([k/2] + 1)^2\rho _1$, where $[x]$ denotes the largest integer less than or equal to $x$.\end{corollary} \noindent {\bf Proof}: Since $\beta \in R$ the arguments of the proof of Lemma~\ref{lm:3} are also valid for the local maxima of $w_n$, which yields the Corollary. \section{Proof of Theorem 1.1} Let $m \leq k$ be a positive integer. By Lemma 2.1 there exists $T:=T(m)$ such that if $t < T$ then $\mu _k < 1$. From general bifurcation theory for simple eigenvalues (see [6]) it follows that there exist two unbounded branches (connected components) of nontrivial solutions bifurcating from $(\mu _m,0)$. We will denote these branches by $G_{m,+}$ and $G_{m,-}$ respectively. In addition, the branch $G_{m,+}$ (respect. $G_{m,-})$ is made up of elements of the form $(\lambda ,w)$, $w$ has $m$ zeros in (0,1], $w(0) > 0$ (respect. $w(0) < 0)$, and contains elements of the form $(\lambda ,w)$ with $\lambda$ near $\mu _m$ and $w(0)$ near zero. Hence $$G_{j,\sigma } \cap G_{\kappa ,s} = \Phi \mbox{ if }(j,\sigma ) \neq (k,s). \eqno{(3.1)}$$ Since $G_{m,s}$, $s\in\{+, -\})$ is unbounded, and since there is no element of $G_{m,s}$ with $\lambda = 0$ (the only solution to (\ref{2.4})--(\ref{2.6}) when $\lambda = 0$ is $w \equiv 0$), Lemma \ref{lm:3} implies that for $m \in \{j, ... ,k\}$ the set $G_{m,s}$ contains an element of the form $(\lambda, w)$ with $\lambda > 1$. By the connectedness of $G_{m,s}$ we see that it contains an element of the form $(1, w_{m,s})$ which proves that $U + w_{m,s}$ is a solution to $(1.1)-(1.2)$. Thus (\ref{1.1})--(\ref{1.2}) has $2(k - j)$ solutions. In addition, since $U(0) > 0$ and $w_{m,+} > 0$ we see that $k - j$ of these solutions are positive at zero, which proves the Theorem. \noindent {\bf Acknowledgement:} The authors wish to thank the referees for their careful reading of the manuscript and constructive suggestions. \begin{thebibliography}{99} \bibitem{} A. Ambrosetti and G. Prodi, {\it On the inversion of some differentiable mappings with singularities between Banach spaces,} Ann. Mat. Pura Appl. 93(1972), 231-247. \bibitem{} A. Castro and A. Kurepa, {\it Energy analysis of a nonlinear singular differential equation and applications}, Rev. Colombiana Mat., 21(1987), 155-166. \bibitem{} A. Castro and A. Kurepa, {\it Radially symmetric solutions to a superlinear Dirichlet problem with jumping nonlinearities}, Trans. Amer. Math. Soc., 315(1989), 353-372. \bibitem{} A. Castro and R. Shivaji, {\it Non-negative solutions for a class of radially symmetric nonpositone problems}, Proc. Amer. Math. 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McKenna, {\it On a conjecture related to the number of solution of a nonlinear Dirichlet problem}, Proc. Roy. Soc. Edin. 95A(1983) 275-283. \bibitem{} A. C. Lazer and P. J. McKenna, {\it Large amplitude periodic oscillations in suspension bridges: some new connections with Nonlinear Analysis}, SIAM Review 32(1990), 537-578. \bibitem{} J. C. de Padua, {\it Multiplicity results for a superlinear Dirichlet problem}, J. Differential Equations, 82(1989), 356-371. \bibitem{} K. Schmitt, {\it Boundary value problems with jumping nonlinearities}, Rocky Mountain J. of Math., 16(1986), 481-495. \end{thebibliography} {\sc Department of Mathematics, University of North Texas, Denton TX 76203} \newline E-mail acastro\@@unt.edu \end{document}