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\markboth{\hfil A sign-changing solution \hfil EJDE--2003/Conf/10}
{EJDE--2003/Conf/10 \hfil A. Castro, P. Dr\'abek, \& J. M. Neuberger\hfil}
\begin{document}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Fifth Mississippi State Conference on Differential Equations and
Computational Simulations, \newline
Electronic Journal of Differential Equations,
Conference 10, 2003, pp 101--107. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
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%
A sign-changing solution for a superlinear Dirichlet problem, II
%
\thanks{ {\em Mathematics Subject Classifications:} 35J20, 35J25, 35J60.
\hfil\break\indent
{\em Key words:} Dirichlet problem, superlinear, subcritical,
sign-changing solution, \hfil\break\indent deformation lemma.
\hfil\break\indent
\copyright 2003 Southwest Texas State University. \hfil\break\indent
Published February 28, 2003.
\hfil\break\indent
P. Drabek was partially supported by
Ministry of Education of the Czech Republic, \hfil\break\indent
MSM 235200001.
\hfil\break\indent
J. Neuberger was partially supported by
the National Science Foundation DMS-0074326.
} }
\date{}
\author{Alfonso Castro, Pavel Dr\'abek, \& John M. Neuberger}
\maketitle
\begin{abstract}
In previous work by Castro, Cossio, and Neuberger \cite{ccn},
it was shown that a superlinear Dirichlet problem has
at least three nontrivial solutions when the derivative of the
nonlinearity at zero is less than the first eigenvalue of
$-\Delta$ with zero Dirichlet boundry condition.
One of these solutions changes sign exactly-once and the other
two are of one sign.
In this paper we show that when this derivative is
between the $k$-th and $k+1$-st eigenvalues there still
exists a solution which changes sign at most $k$ times.
In particular, when $k=1$ the sign-changing {\it exactly-once}
solution persists although one-sign solutions no longer exist.
\end{abstract}
\newtheorem{theorem}{Theorem}
\newtheorem{remark}{Remark}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\numberwithin{equation}{section}
\section{Introduction}
Let $\Omega$ be a smooth bounded region in
$\mathbb{R}^N$, $\Delta$ the Laplacian operator,
and $f\in C^1(\mathbb{R},\mathbb{R})$ such that $f(0)=0$.
In this paper we study
the boundary-value problem
\begin{equation} \label{pde}
\begin{gathered}
\Delta u + f(u) = 0 \quad \hbox{in } \Omega \\
u=0 \quad \hbox{in } {\partial \Omega}.
\end{gathered}
\end{equation}
We assume that there exist constants $A>0$ and
$p \in (1,\frac{N+2}{N-2})$
such that $|f'(u)| \leq A (|u|^{p-1}+1)$ for all $u \in \mathbb{R}$.
Hence $f$ is \textit{subcritical},
i.e., there exists $B>0$ such that $|f(u)| \leq B (|u|^p+1)$.
Also, we assume that there exists $m\in(0,1)$ and $\eta >0$
such that
\begin{equation} \label{crcv_condit}
m u f(u) \geq 2 F(u)
\end{equation}
for $|u| > \eta$, where $F(u) = \int_0^u f(s)\,ds$.
Finally, we make the assumption that $f$ satisfies
\begin{equation}
\label{sprlin_condit}
f'(u) > \frac{f(u)}{u} \quad\hbox{for} \ u \not= 0, \ \
\hbox{and} \ \ \lim_{|u| \to \infty} \frac{f(u)}{u} = \infty
\ \ (f \quad\hbox{is superlinear}).
\end{equation}
Let $H$ be the Sobolev space $H_0^{1,2}(\Omega)$
with inner product
$\langle u,v\rangle=\int_\Omega\nabla u\cdot\nabla v \,d \zeta$
(see \cite{adams} or \cite{GT}).
Let $0 < {\lambda}_1 < {\lambda}_2 \leq {\lambda}_3\leq \cdots$ be
the eigenvalues of $-\Delta$ with zero Dirichlet
boundary condition in $\Omega$. We let
$\{\phi_1, \phi_2, \dots \}$ denote a complete orthonormal
set in $L^2(\Omega)$
of eigenfunctions corresponding to the latter eigenvalues.
Our main result is as follows.
\begin{theorem} \label{main}
If $f'(0)\in[\lambda_k,\lambda_{k+1})$ then {\rm (\ref{pde})} has a
solution $w$
which changes sign at most $k$ times, i.e.,
$\Omega - w^{-1}\{0\}$ consists of at most $k+1$ non-empty connected sets.
\end{theorem}
\begin{corollary} \label{main1}
If $f'(0)\in[\lambda_1,\lambda_{2})$ then {\rm (\ref{pde})} has a solution $w$
which changes sign exactly once.
\end{corollary}
Our proofs here combine Lyapunov-Schmidt reduction arguments \cite{cl},
the mountain pass lemma \cite{rab}, Sard's Lemma \cite{smale},
and the index of critical points of mountain pass
type \cite{hofer}.
To the best of our knowledge, \cite{ccn} was the first to establish the
existence of a {\it sign-changing} solution to (\ref{pde}) for a general
region in the superlinear case where $f'(0)<\lambda_1$.
The proofs in \cite{ccn} are
based on the study of the Nehari manifold
$$
S=\big\{u \in H: u\not = 0, \int_{\Omega}(\|\nabla u\|^2 - uf(u)) \,d \zeta = 0
\big\}.
$$
Unlike the work in \cite{ccn}, where $S$ is a differentible manifold
homeomorphic to the unit sphere and bounded away from $0$,
here $0$ is a limit point of $S$.
Also $S \cup {0}$ has a singularity at $0$. The intersection of
$S$ with planes spanned by $\{\phi_1, \phi_k\}$, $k = 2, \dots $
is a figure eight.
The semipostione result in \cite{semipos} is another example where
a more complicated variational structure is successfully analyzed via
our techniques.
For historical remarks concerning the existence of sign changing
solutions to semilinear elliptic boundary value problems we refer
the reader to \cite{ccn3}. See also \cite{wang}.
\begin{remark} \label{one_sign} \rm
One can easily see that when $f'(0)\geq\lambda_1$ there can be no one
signed solutions.
In fact suppose to the contrary that $f'(0)\geq\lambda_1$ and that $u$ is
(for example) a positive solution.
Let $\phi_1$ be a positive eigenfunction corresponding to $\lambda_1$.
Then, by multiplying (\ref{pde}) by $\phi_1$ and integrating we obtain
\begin{equation} \label{one_sign_eq}
\begin{aligned}
\int_\Omega \{\Delta u + f(u)\}\phi_1 \,d \zeta &=
\int_\Omega \{u \Delta \phi_1 + f(u)\phi_1\} \,d \zeta \\
&= \int_\Omega \{\frac{f(u)}{u}-\lambda_1\}u\phi_1 \,d \zeta \\
&= \int_\Omega \{f'(v) -\lambda_1\}u\phi_1 \,d \zeta\\
&> \int_\Omega \{f'(0)-\lambda_1\}u\phi_1 \,d \zeta \geq 0,
\end{aligned}
\end{equation}
where we have used the mean value theorem to find $v\in(0,u)$.
\end{remark}
\begin{remark} \label{otherbc} \rm
Theorem \ref{main} and Corollary \ref{main1} are also valid when the
Dirichlet boundary condition
in $(\ref{pde})$ is replaced by a homogeneous boundary
condition for which the spectrum of
the Laplacian operator
consists of isolated eigenvalues of finite multiplicity
converging to $\infty$. This is the case, for example, of the
Neumann boundary condition $(\partial u/\partial \eta)(x)=0$
for region with Lipschitzian boundary.
\end{remark}
\section{Preliminary Lemmas}
Let $k$ be a positive integer and
$f'(0) \in (\lambda_k, \lambda_{k+1})$. We define $J:H\to\mathbb{R}$ by
$$
J(u)=\int_\Omega \{\frac12|\nabla u|^2 - F(u) \} \,d \zeta.
$$
By regularity theory for elliptic boundary value problems \cite{GT},
$u$ is a solution to
(\ref{pde}) if and only if $u$ is a critical point of $J$. Because $f$
is subcritical, $J \in C^2(H,\mathbb{R})$ (see \cite{rab}).
The gradient and Hessian of $J$ are given by
\begin{equation} \label{gradJ}
J'(u)(v) =
\langle \nabla J(u), v \rangle = \int_\Omega \{\nabla u \cdot \nabla v - f(u)
v\} \,d \zeta, \quad \hbox{for all } v \in H,
\end{equation}
and
\begin{equation} \label{hessianJ}
\langle D^2J(u)v,w \rangle = \int_\Omega \{\nabla v \cdot \nabla w - f'(u) vw\} \,d \zeta,
\quad \hbox{for all } u,v,w \in H.
\end{equation}
Let $X$ be the linear subspace generated by $\{\phi_1,
\dots, \phi_k\}$ and $Y$ the subspace of $H$ generated by
$\{\phi_{k+1}, \dots \}$. By orthogonality properties of eigenfunctions
$H = X \oplus Y$.
Since (\ref{sprlin_condit}) implies that $f'(t) \geq f'(0) >
\lambda_k$, one sees that there exists $m_1 > 0$ such that
\begin{equation}
\label{negdef}
\langle D^2 J(u)x, x \rangle \leq -m_1 \|x\|^2 \quad\hbox{for all }
u \in H, \; x \in X.
\end{equation}
Arguing as in Theorem 4 of \cite{cl}, one sees
that there exists a function $\psi \in C^1(Y,X)$
such that
\begin{equation}
\label{defjhat}
\hat J(y) \equiv J(y+ \psi(y)) = \max_{x \in X} J(x+y)
\quad\hbox{for all } y\in Y,
\end{equation}
and
\begin{equation}
\label{defjhat1}
\langle \nabla \hat J(y), v \rangle = \langle \nabla J(y+ \psi(y)),
v \rangle = \int_\Omega \{\nabla y \cdot \nabla v - f(y + \psi(y))v\} \,d \zeta,
\end{equation}
for all $ y,v\in Y$.
In addition,
\begin{equation}
\label{defjhat2}
\psi(y) \ \hbox{is the only critical point of} \
\ x \to J(x+ y) \quad\hbox{ for all } \ \ y\in Y.
\end{equation}
Thus
\begin{equation}
\label{reduction}
\hbox{$y$ is a critical point of $\hat J$
if and only if $y + \psi(y)$ is a critical point of $J$.}
\end{equation}
Although not obvious (see \cite{cl}), $\hat J$ is of class $C^2$ in spite
of only
$\psi \in C^1(Y,X)$.
From (\ref{defjhat1}) we see that $\nabla \hat J(y) = y + K(y)$
where $K$ is a compact function of $y$. Also since
$\nabla \hat J$ is a variational vector field
of class $C^1$ we have
$\dim \ker (\nabla \hat J)'(y)$ = $\dim \ker(D^2 \hat J(y))=
\mathop{\rm codim}(D^2( \hat J(y))(Y)) =\mathop{\rm codim}(\nabla J)'(y)(Y) $
for all $y\in Y$. Thus we may apply Sard's lemma \cite{smale} to
conlcude the following lemma.
\begin{lemma} \label{sards}
There exists $\{q_n\}\subset Y$ with $q_n\downarrow 0$ as
$n\to\infty$ such that if $\nabla \hat J(u) =q_n$
then
the Hessian $D^2\hat J(u)$ is invertible.
\end{lemma}
\paragraph{Proof}
This proof follows immediately from the fact that the regular
values of $\nabla \hat J$ is the complement of a set of first
category. In partiuclar it is dense. \hfill$\diamondsuit$
Let $q_n$ be as in the previous lemma. We define $J_n: Y \to \mathbb{R}$ by
$J_n(y) = \hat J(y) - \langle q_n, y \rangle$. We note that
$D^2J_n(y) = D^2\hat J (y)$ for each $y \in Y$. Also from
(12) of \cite{cl}
\begin{equation}
\label{d2jn}
\langle D^2 \hat J(y)h,h \rangle =
\langle D^2 J(y + \psi(y))(h+\psi'(y)h), h+\psi'(y)h \rangle.
\end{equation}
\begin{lemma}
\label{mountain}
The functional $J_n$ has a critical point $y_n$
such that the Morse index of $D^2J(y_n+\psi(y_n))$
is less than or equal to $k+1$.
\end{lemma}
\paragraph{proof}
In order to establish the existence of the critical point $u_n$
we prove that $J_n$ satisifies the hypotheses of the Mountain
Pass Lemma \cite{rab}.
Since $f'(0) < \lambda_{k+1}$, we have
$\langle D^2 J(0)y,y\rangle = J''(0)(y,y)\geq(1-f'(0)/\lambda_{k+1})\int_\Omega
\ |\nabla y |^2 \,d \zeta>0$ for $y \in Y$. Thus $0$ is a strict local
minimum
of $J$ restricted to $Y$. Thus there exist
$\delta, \eta>0$
such that $J(y)\geq \eta$ for all $||y||=\delta$.
This and (\ref{defjhat}) imply that
for $\|y\| = \delta$ we have
\begin{equation}
\label{mpl1}
\hat J(y) \geq J(y) \geq \eta > 0\,.
\end{equation}
Now taking $n$ sufficiently large so that $\|q_n\| \leq
\eta/(2 \delta)$ we have
\begin{equation}
\label{mpl2}
J_n(y) \geq \eta - \|q_n\|\delta > \eta/2 >0\,,
\end{equation}
for $\|y\| = \delta$.
Next we note that, since $0$ is a critical point of
$J$, $\psi(0)=0$. Hence $J_n(0) = 0$.
Additionally, since we are assuming $f$ to be superlinear,
there exit numbers $m_1 > \lambda_{k+1}$ and $m_2$
such that
\begin{equation}
\label{Fgreat}
2F(t) \geq m_1 t^2 + m_2
\end{equation}
for all $t \in \mathbb{R}$.
Hence
\begin{equation}
\label{mpl3}
\begin{aligned}
J_n(t\phi_{k+1}) & = \hat J(t \phi_{k+1}) - \langle q_n, t\phi_{k+1}
\rangle \leq J(t \phi_{k+1}) - \langle q_n, t\phi_{k+1}
\rangle \\
& \leq (t\|\phi_{k+1}\|^2
- m_1\int_\Omega (t\phi_{k+1})^2 \,d\zeta - m_2 |\Omega|)/2 +
t \| q_n \| \| \phi_{k+1}\| \\
& \quad \to -\infty \quad\hbox{as } t \to +\infty,
\end{aligned}
\end{equation}
where we have used that $\|\phi_{k+1}\|^2
= \lambda_{k+1} \int_\Omega \phi_{k+1}^2 \,d \zeta.$
For future reference we note that, without loss of generality, we may
assume that $\|q_n\| \leq 1$ for all $n$. Thus (\ref{mpl3})
implies that
\begin{equation}
\label{mplK}
J_n(t\phi_{k+1}) \leq \frac{(\lambda_{k+1} - m_1)(-m_2|\Omega|)-
\lambda_{k+1} \|\phi_{k+1}\|}{2(\lambda_{k+1} - m_1)} \equiv
\tilde K \quad\hbox{for all } n, t \geq 0.
\end{equation}
From (\ref{mpl3}), for
each $n$, there exists a real number $t_n > 0$ with
$\|t_n \phi_{k+1}\| \geq 2\delta $
such that
$\hat J_n(t_n \phi_{k+1}) < 0$.
Next we show that $J_n$ satisfies the Palais-Smale condition.
Suppose that $\{y_j\}$ is a sequence so that
$\{J_n(y_j)\}$ is bounded, say $|J_n(y_j) |\leq M$ for
all $j$ and $\nabla J_n(y_j)\to 0$ as $j\to\infty$.
For ease of notation, let
$u=y_j+\psi(y_j)$ and $T=\frac{m}{2}uf(u)-F(u)$.
Then
\begin{equation}
\label{ps}
\begin{aligned}
M + \frac{m}{2}||u|| & \geq J_n(y_j)-\frac{m}{2}[\langle \nabla J_n
(y_j), y_j \rangle + \langle \nabla J(y_j + \psi(y_j)), \psi(y_j) \rangle] \\
& = (\frac12 - \frac{m}{2})||u||^2 + \int_\Omega T \,d \zeta -
(1-\frac{m}{2})\langle q_n, u \rangle \\
& \geq (\frac14 - \frac{m}{4})(||u||^2 -\|q_n\|^2) + M_1|\Omega|,
\end{aligned}
\end{equation}
where $M_1 \in \mathbb{R}$ is a lower bound for $T$
(see (\ref{crcv_condit})).
The latter inequality implies that
$\{y_j+\psi(y_j)\}$ is bounded. Hence without loss of generality
we may assume that $\{y_j\}$ converges weakly to $\bar y \in Y$
and that $\psi(y_j)$ converges to $\bar x \in X$. Since the
imbedding of $H$ in $L^{p+1}(\Omega)$ is compact, we may assume
that $f(y_j + \psi(y_j))$ converges strongly in $L^1(\Omega)$.
Since $\nabla J_n(y_j) = y_j + K(y_j) + q_n \to 0$ with $K$ compact,
we
see that $\{y_j\}$ has a convergent subsequence. This proves that
$J_n$ satisfies the Palais-Smale condition.
Now by the Mountain Pass Lemma there exists $y_n \in Y$ such that
$\nabla J_n(y_n) = 0$ and
\begin{equation}
\label{mpl5}
J_n(y_n) = \inf_{\sigma \in \Sigma} \ \big[ \max_{t \in [0,1]}
J_n(\sigma (t)) \big],
\end{equation}
where $\Sigma = \{\sigma:[0,1] \to Y; \sigma$ is continuous,
$\sigma(0)=0$ and $\sigma(1) = t_n \phi_{k+1}\}$.
Thus from (\ref{mpl2}), (\ref{mplK}) and (\ref{mpl5}) we have
\begin{equation}
\label{mpl6}
\eta/2 \leq J_n(y_n) \leq \tilde K.
\end{equation}
In addition, since
$D^2J_n(y_n)$ is invertible (see Lemma \ref{sards}), by Theorem 2 of
\cite{hofer},
we may assume that the Morse index of $D^2J_n(y_n)$ is 1.
Since $D^2J(y_n + \psi(y_n))$ is negative definite in a
subspace of dimension $k$, namely $X$, then we conclude that
the Morse index of $D^2J(y_n + \psi(y_n))$ is at most
$k+1$, and this concludes the proof. \hfill$\diamondsuit$
\section{Proof of Main Theorem}
First we consider the case $f'(0) > \lambda_k$.
Let $\{y_n\}$ be as in
Lemma \ref{mountain}. By (\ref{mpl6})
one sees that
$\{y_n+\psi(y_n)\}$ is bounded. Thus, without loss of
generality, we may assume that $\{y_n\}$ converges
weakly to $\bar y \in Y$ and $\{\psi(y_n)\}$ converges to
$\bar x \in X$.
From (\ref{defjhat}) we have
$0 = \nabla J_n(y_n) = y_n + K(y_n) - q_n$ where $K$
is a compact operator. Since, in addition $\{q_n\}$
converges
to $0$, actually $\{y_n\}$ converges strongly to $\bar y$.
Also since $J_n(y_n) \geq \eta/2 > 0$, we see that $\hat J
(\bar y) \geq \eta/2 > 0$ and $\bar y \not = 0$.
Now for $v \in
X$ one has
\begin{equation}\label{teo1}
\begin{aligned}
&\langle \bar x, v \rangle - \int_\Omega \{ f(\bar y + \bar x)v\} \,d \zeta\\
& = \lim_{n \to \infty}
\big[\langle \psi(y_n), v \rangle - \int_\Omega \{v f(y_n
+ \psi(y_n))\} \,d \zeta - \langle q_n, \psi(y_n) \rangle\big] \\
&= 0.
\end{aligned}
\end{equation}
Hence $\bar x = \psi(\bar y)$ and $\nabla J(\bar x + \bar y) = 0$.
Thus $\bar x + \bar y \not = 0$ is a solution to (\ref{pde}). Let us see
that $\bar x + \bar y$ has at most $k+1$ nodal regions.
If not, by defining $v_j$, $j = 1, \dots , k+2$, as $\bar x + \bar y$
on $W_j$ and as zero on $\bar \Omega - W_j$,
then from (\ref{sprlin_condit}),
(\ref{gradJ}) and (\ref{hessianJ}),
we
see that
$\langle D^2J(\bar x + \bar y)v_j, v_j \rangle < 0$.
Since the $v_j's$ are mutually orthogonal then we have
that $D^2J(\bar x + \bar y)$ is negative
definite
on a $k+2$-dimensional subspace. By continuity then $D^2J(y_n +
\psi(y_n))$ is negative definite on the same $k+2$-dimensional
subspace. This contradicts that the Morse index of $D^2J(y_n +
\psi(y_n))$ is less than or equal to $k+1$. This contradiction proves
that $\bar x + \bar y$ is a solution to (\ref{pde}) having at
most $k+1$ nodal regions.
Finally we consider the case $f'(0) = \lambda_k$.
Let $\{\epsilon_j\}$ be a sequence of positive numbers converging
to $0$. Without loss of generality we may assume that
$\epsilon_j < (\lambda_{k+1} - \lambda_k)/2$ for all positive
integers $j$. By our previous arguments, there exists a sequence
$\{ u_j = \bar x_j + \bar y_j\}$ of functions in $H$ that
satisfy
\begin{equation} \label{pdej}
\begin{gathered}
\Delta u_j + \epsilon_j u_j +f(u_j) = 0 \quad \hbox{in } \Omega \\
u_j=0 \quad \hbox{in } \partial \Omega.
\end{gathered}
\end{equation}
In addition, each $x_j + y_j$ is the limit a of a sequence
$\{x_{n,j} + y_{n,j}\}$ with each $x_{n,j} + y_{n,j}$
satisfying (\ref{mpl6}). Hence, by continuity $\eta/2
\leq J(\bar x_j + \bar y_j) \leq \tilde K$ and
$$
\int_\Omega \{\nabla v \cdot \nabla v - (f'(u_j) + \epsilon_j) v^2 \} \,d \zeta
$$
defines a quadratic form of Morse index
at most $k+1$. Arguing as above one sees that
$\bar x_j + \bar y_j$ has a convergent subsequence with limit
$\bar x + \bar y$. The function $\bar x + \bar y$ is
a solution to (\ref{pde}) and, as in the case $f'(0)
> \lambda_k$, it has at most $k+1$ nodal regions.
This concludes the proof of our main theorem;
the corollary is obvious.
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\end{thebibliography}
\noindent\textsc{Alfonso Castro}\\
Division of Mathematics and Statistics,
University of Texas at San Antonio, \\
San Antonio, TX 78249-0664, USA\\
E-mail: acastro@utsa.edu \smallskip
\noindent\textsc{Pavel Drabek}\\
Department of Mathematics,
University of West Bohemia, \\
306 14 Pilsen, Czech Republic.\\
E-mail address: pdrabek@kma.zcu.cz \smallskip
\noindent\textsc{John M. Neuberger}\\
Department of Mathematics,
Northern Arizona University,\\
Flagstaff, AZ 86011-5717 USA.\\
E-mail address: John.Neuberger@nau.edu
\end{document}