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\markboth{\hfil THE LAZER McKENNA CONJECTURE \hfil
EJDE--1993/07}%
{EJDE--1993/07\hfil Alfonso Castro and Sudhasree Gadam \hfil}
\begin{document}
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\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
\begin{equation}
\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}\end{equation}
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
\begin{equation} \alpha < \rho_1([j/2] + 1)^2 < \rho_k < \beta <
\rho_{k+1}\label{1.5}\label{1.4}\end{equation}
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
\begin{equation} \mu_j \leq \sup_{u\in M-\{0\}}
({\int}_\Omega \ \nabla u\cdot \nabla u )/({\int}_\Omega g'(U) u^2 )\,,
\label{2.10}\end{equation}
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
\begin{equation} E(r,t,\lambda ,d)\rightarrow
+\infty\mbox{ uniformly on [0,1]
as $|d|$ tends to
infinity.}\label{2.11}\end{equation}
\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.
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\end{thebibliography}
{\sc Department of Mathematics, University of North Texas, Denton TX 76203}
\newline E-mail acastro\@@unt.edu
\end{document}