\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 11, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2006/11\hfil On a variational approach]
{On a variational approach to existence and multiplicity results
for semipositone problems}
\author[D. G. Costa, H. Tehrani, J. Yang\hfil EJDE-2006/11\hfilneg]
{David G. Costa, Hossein Tehrani, Jianfu Yang} % in alphabetical order
\address{David G. Costa \hfill\break
Dept. of Mathematical Sciences \\
University of Nevada - Las Vegas \\
Las Vegas, NV 89154-4020, USA}
\email{costa@unlv.nevada.edu}
\address{Hossein Tehrani \hfill\break
Dept. of Mathematical Sciences \\
University of Nevada - Las Vegas \\
Las Vegas, NV 89154-4020, USA}
\email{tehranih@unlv.nevada.edu}
\address{Jianfu Yang \hfill\break
Wuhan Institute of Physics and Mathematics \\
Chinese Academy of Sciences \\
P.O. Box 71010, Wuhan 430071, China}
\email{jfyang@wipm.ac.cn}
\date{}
\thanks{Submitted June 15, 2005. Published January 24, 2006.}
\thanks{J. Yang was supported by grant 10571175 from the NSF of
China.}
\subjclass[2000]{46E35, 46E39, 35J55}
\keywords{Variational approach; semipositone problems}
\begin{abstract}
In this paper we present a novel variational approach to the
question of existence and multiplicity of positive solutions to
semipositone problems in a bounded smooth domain of
$\mathbb{R}^N$. We consider both the sublinear and superlinear
cases.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
Let $\Omega\subset\mathbb{R}^N$ be a smooth bounded domain. We are
interested in presenting a variational approach to the question of
finding {\it positive solutions} (i.e. nonnegative solutions
without interior zeros in $\Omega$) to a class of problems of the
form
\begin{equation} \label{Pl}
\left\{\begin{gathered}
- \Delta u = \lambda f(u)\quad\text{in }\Omega \\
u = 0\quad\text{on }\partial\Omega,
\end{gathered}\right.
\end{equation}
where $\lambda$ is a positive parameter and $f : [0,+\infty)
\to \mathbb{R}$ is a continuous function satisfying
the condition
\begin{itemize}
\item[(F0)] $f(0) = -a < 0$.
\end{itemize}
Such problems are usually referred in the literature as {\it
semipositone problems}. We refer the reader to \cite{castroS0},
where Castro and Shivaji initially called them {\it nonpositone
problems}, in contrast with the terminology {\it positone
problems}, coined by Cohen and Keller in \cite{cohenK}, when the
nonlinearity $f$ was positive and monotone. Here we
will consider both the {\it sublinear case}, where $f$ satisfies
\begin{itemize}
\item[(F1)] $\lim_{s\to +\infty}\frac{f(s)}{s} = 0 < \lambda_1$,
\end{itemize}
(with $\lambda_1 > 0$ denoting the first eigenvalue of $-\Delta$
under Dirichlet boundary condition on $\Omega$) and the {\it
superlinear, subcritical case}, where $f$ is such that
\begin{itemize}
\item[(F2)] $\lim_{s\to +\infty}\frac{f(s)}{s} = +\infty$,
\quad $|f(s)|\leq C(1+|s|^{p-2})$,
\end{itemize}
with $2\leq p < 2^{\ast}=\frac{2N}{N-2}$ if $N\geq 3$
($2^{\ast}=+\infty$ if $N=1,2$). In this latter case, an
assumption that is usually made to deal with compactness
properties is the {\it Ambrosetti-Rabinowitz} condition:
\begin{itemize}
\item[(\^F2)] $F(s) \leq \theta f(s) s$ for all
$s\geq K$ (and some $\theta\in (0,\frac{1}{2}$)).
\end{itemize}
The usual approaches to such semipositone problems are through
{\it quadrature methods} (see e.g. \cite{castroS, castroHS}), the
method of {\it sub-super-solution} (e.g. \cite{anuradhaDS,
castroGS}), {\it degree theory} and/or {\it bifurcation theory}
(see e.g. \cite{allegrettoNZ, ambrosettiAB}). We refer the author
to the survey paper by Castro-Maya-Shivaji \cite{castroMS} and
references therein. Let us consider the sublinear case, for
example. As is well-known, in this case a super-solution can be
easily found by considering the solution $\overline{u} > 0$ of the
linear problem
$$
- \Delta u = \lambda (\epsilon u + B)\, ,
$$
where $0 < \lambda\epsilon < \lambda_1$ and $B > 0$ are such that
(cf. (F1))
$$
f(s) \leq \epsilon s + B\quad\ \forall\ s > 0\,.
$$
Moreover, by using the maximum principle, it follows that such a
super-solution is an upper bound for any positive solution, or
even sub-solution, of \eqref{Pl} (see \cite{figueiredo}).
Therefore, the main difficulty in proving the existence of a
positive solution for \eqref{Pl} consists in finding a
positive sub-solution. As a matter of fact, as can be easily seen,
no positive sub-solution can exist if $f$ does not assume positive
values; and the fact that $f$ has negative values for $s > 0$
small precludes the existence of any such sub-solution with small
$L^{\infty}$ norm. Thus the nonlinearity $f$ must assume positive
values and, as suggested by the results in \cite{clementS}, that
should happen in such a way that
\begin{itemize}
\item[(F3)] $F(\delta) > 0$ for some $\delta > 0$,
\end{itemize}
where $F(u)=\int_{0}^{u} f(s)ds$ as usual. In addition, one must
also have $\lambda$ bounded away from zero (see \cite{mayaS} and
Lemma \ref{lemma1} below).
As already mentioned, our main objective in this article is to
present a variational approach to the question of existence and
multiplicity of positive solutions to such semipositone problems.
We will do so by looking at \eqref{Pl} as a problem with the
discontinuous nonlinearity $g(s)$ defined by
\begin{equation}\label{eq0}
g(s)= H(s) f(s) = \begin{cases}
0 & \mbox{if } s\leq 0 \\
f(s) & \mbox{if } s > 0\,,
\end{cases}
\end{equation}
where $H(s)=0$ for $s\leq 0$, $H(s)=1$ for $s > 0$ denotes the
Heaviside function. More precisely, we will be considering the
slightly modified problem
\begin{equation} \label{hPl}
\left\{\begin{gathered}
- \Delta u = \lambda g(u) \quad\text{in }\Omega \\
u = 0\quad\text{on }\partial\Omega\,.
\end{gathered}
\right.
\end{equation}
We note that the set of positive solutions of
\eqref{Pl} and \eqref{hPl} do coincide.
Moreover, any non-zero solution $u$ of \eqref{hPl}
is nonnegative and, in fact, if the set $A_u := \{ x\in\Omega\ |\
u(x)=0\}$ has measure zero then $u$ is an \mbox{{\it a.e.
positive}} solution of \eqref{Pl}. We will show this to be
the case for some solutions when $\Omega$ is a ball.
We should mention that our results were inspired by the works
of Ambrosetti-Struwe \cite{ambrosettiS} and Chang \cite{chang}.
On the other hand, we are not aware of any other
work where solutions of semipositone problems were obtained
directly through variational techniques. However, the authors in
\cite{arcoyaC} have considered existence results for problem
\eqref{hPl} through approximation of the
discontinuous nonlinearity by a sequence of continuous functions.
Variational methods were then applied to the corresponding
sequence of problems and limits were taken. We believe that our
direct variational approach to such problems is rather natural and
conducive to dealing with more general situations.
Our main results concerning problems \eqref{Pl} and
\eqref{hPl} are the following:
\begin{theorem}\label{thm0.1}
Assume \emph{(F0), (F1)} and \emph{(F3)}. Then, there exist
$0 < \Lambda_0\leq\Lambda_2$ such that \eqref{hPl} has no
nontrivial nonnegative solution for $0< \lambda < \Lambda_0$, and
has at least two nontrivial nonnegative solutions $\widehat{u}_{\lambda}$,
$\widehat{v}_{\lambda}$ for all $\lambda > \Lambda_2$. Moreover, when
$\Omega$ is a ball $B_R = B_R (0)$, these two solutions are
non-increasing, radially symmetric and, if $N\geq 2$, at least one
of them is positive, hence a solution of \eqref{Pl}.
\end{theorem}
\begin{theorem}\label{thm0.2}
Assume \emph{(F0), (F2), (\^F2)} and \emph{(F3)}. Then,
\eqref{hPl} has at least one nonnegative solution
$\widehat{v}_{\lambda}$ for all $\lambda > 0$. If $\Omega = B_R$ then
$\widehat{v}_{\lambda}$ is non-increasing, radially symmetric and one of
the two alternatives occurs:
\begin{itemize}
\item[(i)] There exists $\Lambda_1 > 0$ such that, for all $0 <
\lambda < \Lambda_1$, $\widehat{v}_{\lambda}$ is a positive solution
of \eqref{Pl} having negative normal derivative on
$\partial B_R$;
\item[(ii)] For some sequence $\mu_n \to 0$, problem
$(P_{\mu_n})$ has a positive solution $\widehat{w}_{\mu_n}$ with
zero normal derivative on $\partial B_R$.
\end{itemize}
\end{theorem}
\section{The Abstract Framework}
We start by recalling some basic results on variational methods
for locally Lipschitz functionals $I : X \to
\mathbb{R}$ defined on a real Banach space $X$ with norm
$\|\cdot\|$ (cf. \cite{clarke,chang}), that is, for functionals
such that, for each $u\in X$, there is a neighborhood $N=N_u$ of
$u$ and a constant $K=K_u$ for which
$$
|I(v)-I(w)|\leq K \| v-w\|\quad\forall v,w\in N.
$$
For given $u, h\in X$, the {\it generalized directional derivative of}
$I$ {\it at} $u$ {\it in the direction of} $h$ is defined by the
formula
$$
I^{0}(u;h) = \limsup_{k\to 0,\, t\downarrow 0}\frac{1}{t}[ I(u
+ k + t h) - I(u + k) ]
$$
The following properties are known to hold:
\begin{itemize}
\item[$(i)$] $h\mapsto I^{0}(u;h)$ is sub-additive, positively
homogeneous, continuous, and convex;
\item[$(ii)$] $|I^{0}(u;h)|\leq K_u \| h\|$;
\item[$(iii)$] $I^{0}(u;-h) = (-I)^{0}(u;-h)$.
\end{itemize}
Therefore, the so-called {\it generalized gradient of} $I$ {\it
at} $u$, written $\partial I (u)$, is defined as the
subdifferential of the convex function $I^{0}(u;h)$ at $h=0$, that
is,
$$
\mu\in\partial I (u) \subset X^{\ast}\quad \Longleftrightarrow \quad
\langle \mu , h\rangle \leq I^{0}(u;h)\quad \forall h\in X.
$$
For the convenience of the reader, we list below some of the main
properties of the generalized gradient $\partial I (u)$:
\begin{enumerate}
\item%[(P1)]
For each $u\in X$, $\partial I (u)$ is a non-empty
convex and $w^{\ast}$-compact subset of $X^{\ast}$;
\item%[(P2)]
$\|\mu\|_{X^{\ast}}\leq K_u$ for all $\mu\in\partial I (u)$;
\item%[(P3)]
If $I,\ J : X \to \mathbb{R}$ are locally
Lipschitz functionals then
$$
\partial (I + J)(u) \subset \partial I (u) + \partial J (u);
$$
\item%[(P4)]
$\partial (\lambda I) (u) = \lambda \partial I (u)$ for all
$\lambda\in\mathbb{R}$;
\item%[(P5)]
If $I$ is a convex functional then $\partial I (u)$
coincides with the usual subdifferential of $I$ in the sense of
convex analysis;
\item%[(P6)]
If $I$ has a Gateaux derivative $DI(v)$ at every point $v$ of
a neighborhood $N$ of $u$ and $DI : N \to X^{\ast}$ is
continuous, then $\partial I(u) = \{ DI(u)\}$;
\item%[(P7)]
$I^{0}(u;h) =
\max\{ \langle \mu , h\rangle\ |\ \mu\in\partial I(u)\}$ for
all $h\in X$;
\item%[(P8)]
If $I$ has a local minimum (or a local maximum) at
$u_0 \in X$ then $0 \in \partial I(u_0)$.
\end{enumerate}
Now, by definition, one says that $u\in X$ is a {\it critical
point} of the locally Lipschitz functional $I$ if
$$
0\in\partial I(u).
$$
In this case the real number $c=I(u)$ is called a critical value
of $I$. Note that property (8) above says that a local minimum
(or local maximum) of $I$ is a critical point of $I$.
On the other hand, $I$ is said to satisfy the Palais-Smale
condition $(PS)_c$ at the level $c\in\mathbb{R}$ if, for any
sequence $(u_n)$ such that $I(u_n)\to c$ and
$\lambda (u_n):=\min_{\mu\in\partial I(u_n)}\|\mu\|_{X^{\ast}}\to 0$,
one can extract a convergent
subsequence. Finally, we point out that many of the results of the
classical critical point theory have been extended by Chang
\cite{chang} to this setting of locally Lipschitz functionals. For
example, one has the celebrated:
\begin{theorem}[Mountain-Pass Theorem, Ambrosetti-Rabinowitz
\cite{ambrosettiR}]
Let $X$ be a reflexive Banach space and
$I:X\to\mathbb{R}$ be a locally Lipschitz functional
satisfying $(PS)_c$ for all $c > 0$ and the following geometric
conditions:
\begin{itemize}
\item[$(i)$] $I(0)=0$ and there exist $\rho ,\alpha > 0$ such that
$I(u)\geq\alpha$ if $\| u\| = \rho$;
\item[$(ii)$] there exists $e\in X$ such that $\| e\| > \rho$ and
$I(e)\leq 0$.
\end{itemize}
Then $I$ has a critical value $c\geq \alpha$ given by
$$
c = \inf_{\gamma\in\Gamma}\sup_{t\in [0,1]}I(\gamma (t))\,,
$$
where $\Gamma = \{ \gamma\in C([0,1],X)\ |\ \gamma (0)=0\,,
\gamma (1)=e \}$.
\end{theorem}
For the rest of this article, we denote the $H^1_0$-norm by
$\|u\|=(\int_{\Omega}|\nabla u|^2\,dx)^{1/2}$ and we often use
the same letter $C$ to represent various positive constants.
\section{Proofs of the Main Results}
Now, having listed some basic results on critical point theory for
Lipschitz functionals, let us consider the functional
$$
I_{\lambda}(u) = \frac{1}{2}\int_{\Omega}|\nabla u|^2\,dx -
\lambda \int_{\Omega}G(u)\,dx\, ,
$$
where $G(u)=\int_{0}^{u}g(s)\, ds$ and $g(s)$ were defined in
\eqref{eq0}.
%%%%%%%%%%%%%%%%%(\eqref{e0.1}).
Clearly $G : \mathbb{R}\to \mathbb{R}$ is a locally
Lipschitz continuous function and satisfies $G(s)=0$ for
$s\leq 0$. In view of \cite[Theorems 2.1 and 2.2]{chang}, the above
formula for $I_{\lambda}(u)$ defines a locally Lipschitz
functional on $H^1_0 (\Omega)$ whose critical points are solutions
of the differential inclusion
$$
-\Delta u (x) \in \lambda
[\underline{g}(u(x)),\overline{g}(u(x))]\quad\mbox{a.e. in }\Omega\,,
$$
where $\underline{g}(s):=\min\{ g(s - 0), g(s + 0)\}$ and
$\overline{g}(s):=\max\{ g(s - 0), g(s + 0)\}$. In our present
case, it follows that $\underline{g}(s) = \overline{g}(s) = f(s)$
for $s > 0$, $\underline{g}(s) = \overline{g}(s) = 0$ for $s < 0$,
and $\underline{g}(0) = -a$, $\overline{g}(0) = 0$.
We start with some preliminary lemmas.
\begin{lemma}\label{lemma1}
Assume \emph{(F0), (F1)} and \emph{(F3)}.
Then there exists $\Lambda_0 > 0$ such that \eqref{hPl} has no
nontrivial solution
$0\leq u\in H^1_0 (\Omega)$ for $0 < \lambda < \Lambda_0$.
\end{lemma}
\begin{proof} If $u\geq 0$ is a solution of \eqref{hPl}
then, multiplying the equation by $u$ and integrating over
$\Omega$ yields
$$
\frac{1}{2}\| u\|^2 = \lambda \int_{\Omega}g(u) u\,dx = \lambda
\Big( \int_{[u\leq \delta_0]}u g(u)\,dx + \int_{[u\geq
\delta_0]}u g(u)\,dx\Big)\,,
$$
hence
\begin{equation}\label{e2.1}
\frac{1}{2}\| u\|^2\leq\lambda\int_{[u\geq \delta_0]}u g(u)\,dx\, ,
\end{equation}
where we have chosen $\delta_0 > 0$ so that $g(s)\leq 0$ for
$0\leq s\leq \delta_0$ (such a $\delta_0$ exists in view of
(F0)). Now, since (F1) implies the existence of $C > 0$ such
that
$$
s g(s) \leq C (1+s^2)
$$
for all $s \geq 0$, we obtain from \eqref{e2.1} that
$$
\frac{1}{2}\| u\|^2 \leq \lambda C\int_{[u\geq \delta_0]}(1 +
u^2)\,dx \leq \lambda C (\frac{1}{\delta_{0}^{2}} + 1)
\int_{[u\geq\delta_0]}u^2\,dx\leq\lambda C \int_{\Omega}u^2\,dx\,,
$$
so that
$$
\frac{1}{2}\| u\|^2 \leq \lambda C \| u\|^2\,,
$$
where this last constant $C > 0$ is independent of both $u$ and
$\lambda$. Therefore we must have
$$
\lambda \geq \frac{1}{2C} := \Lambda_0 > 0\,.
$$
\end{proof}
\begin{lemma}\label{lemma2}
Assume \emph{(F0)} and either \emph{(F1)} or \emph{(F2)}.
Then $u=0$ is a
strict local minimum of the functional $I_{\lambda}$.
\end{lemma}
\begin{proof} In fact, we only need to assume (F0) and the condition
$$
G(s)\leq C (1+|s|^{2^{\ast}}) \mbox{ for all $s\in\mathbb{R}$},
$$
which is implied by either (F1) or (F2). Recall also that
$G(s)=0$ for $s\leq 0$. Then, with $\delta_0 > 0$ as in the proof
of Lemma \ref{lemma1} and noticing that $G(s)\leq 0$ for all
$-\infty < s\leq\delta_0$, we can write for an arbitrary
$u\in H^1_0 (\Omega)$,
\begin{align*}
I_{\lambda}(u) &= \frac{1}{2}\| u\|^2 - \lambda
\int_{\Omega}G(u)\,dx \\
&\geq \frac{1}{2}\| u\|^2 - \lambda
\int_{[u\geq \delta_0]}G(u)\,dx\\
&\geq \frac{1}{2}\| u\|^2 - \lambda
C \int_{[u\geq \delta_0]}(1 + u^{2^{\ast}})\,dx\\
&\geq \frac{1}{2}\| u\|^2 - \lambda
C (\frac{1}{\delta_{0}^{2^{\ast}}} + 1)
\int_{[u\geq \delta_0]}u^{2^{\ast}}\,dx\,,
\end{align*}
so that, using Sobolev embedding theorem in the last inequality,
and with a constant $C > 0$ independent of $u$ and $\Omega$, we
obtain
$$
I_{\lambda}(u) \geq \frac{1}{2}\| u\|^2 - \lambda C\|
u\|^{2^{\ast}} = \frac{1}{2} \| u\|^{2}(1 - 2\lambda C \|
u\|^{2^{\ast} - 2}).
$$
Therefore, for each
$0< \rho < \rho_{\lambda}:= 1/(2\lambda C)^{2^{\ast}-2}$,
it follows that $I_{\lambda}(u)\geq \alpha_{\rho} > 0$
if $\| u\| = \rho$. This shows that $u=0$ is a
strict local minimum of $I_{\lambda}$.
\end{proof}
\begin{remark}\label{rem2.3} \rm
We note that both $\rho_{\lambda} > 0$ and $\alpha_{\rho}> 0$
obtained in the above proof do not depend on the domain $\Omega$.
\end{remark}
\begin{lemma}\label{lemma3}
Under the same assumptions as in Lemma \ref{lemma2}, let
$\widehat{u}\in H^1_0 (\Omega)$ be a critical point of
$I_{\lambda}$. Then,
$\widehat{u}\in C^{1,\epsilon}(\overline{\Omega})$ and
$\widehat{u}$ is a nonnegative solution of \eqref{hPl}.
\end{lemma}
\begin{proof} We will follow some of the arguments in
\cite{ambrosettiS,chang}. As mentioned earlier, if
$\widehat{u}$ is a critical point of $I_{\lambda}$ then it
is shown in \cite{chang} that
$\widehat{u}$ is a solution of the differential inclusion
\begin{equation}\label{diffincl}
-\Delta u \in \lambda [\underline{g}(u),\overline{g}(u)]\quad
\mbox{a.e. in} \Omega\,,
\end{equation}
where $\underline{g}(s)=\min\{ g(s - 0), g(s + 0)\}$ and
$\overline{g}(s)=\max\{ g(s - 0), g(s + 0)\}$. Since $g$ is
only discontinuous at $s=0$, the above differential inclusion
reduces to an equality, except possibly on the subset
$A\subset\Omega$ where $\widehat{u}=0$. And, in this latter case,
$-\Delta\widehat{u}$ takes on values in the bounded interval
$[-a,0]$. Therefore, by standard elliptic regularity, it follows that
$\widehat{u}\in H^1_0 \cap W^{2,p}(\Omega)$ for all $p\geq 2$. In
particular, $\widehat{u}$ is of class $C^{1,\epsilon}$,
$0 < \epsilon < 1$.
Next, in view of a well-known result of Stampacchia, we have that
$-\Delta\widehat{u} = 0$ a.e. in $A$. Therefore, since we defined
$g(0) = 0$, it follows that
$$
-\Delta \widehat{u} = g(\widehat{u})\quad \mbox{a.e. in }\Omega.
$$
Replacing the inclusion \eqref{diffincl} on $\widehat{u}$, we
conclude that $\widehat{u}\in C^{1,\epsilon}(\overline{\Omega})$
is a solution of \eqref{hPl}. Finally, recalling
that $g(s)=0$ for $s\leq 0$, it is clear that $\widehat{u}\geq 0$.
The proof of Lemma \ref{lemma3} is complete.
\end{proof}
\begin{lemma}\label{lemma4}
Assume either \emph{(F1)} or \emph{(F2), (\^F2)}. Then
$I_{\lambda}$ satisfies the Palais-Smale condition $(PS)_c$ at
every $c\in\mathbb{R}$.
\end{lemma}
\begin{proof} The proof in either case is a direct consequence of
Theorem 4.3 and Theorem 4.4, respectively, in Chang \cite{chang}. In the
superlinear case, it only suffices to notice that
(\^F2) implies the corresponding condition in
Theorem 4.4,
\begin{equation}\label{e2.2}
J(u) \leq \theta \min_{\mu\in\partial J(u)}\langle \mu ,u\rangle +
M \quad\forall u\in H^1_0 (\Omega)\,,
\end{equation}
where $J(u)=\int_{\Omega}G(u)\,dx$, $u\in H^1_0 (\Omega)$, in our
present case. But this follows immediately by observing that we
can identify $\mu\in (H^1_0)^{\ast}$ with a function $w\in H^1_0$
and that the inclusion
$$
\partial J(u) \subset [\underline{g}(u),\overline{g}(u)]
$$
says that given $w\in\partial J(u)$ then $w(x)=g(u(x))$ if
$u(x)\neq 0$, $w(x)\in [-a,0]$ if $u(x)=0$. Therefore,
$$
\langle w, u\rangle = \int_{\Omega}g(u) u\,dx\quad \mbox{for
all $w\in\partial J(u)$},
$$
so that (\^F2) clearly implies (\eqref{e2.2})
\end{proof}
\begin{lemma}\label{lemma5}
Under assumptions \emph{(F0)} and \emph{(F1)},
let $\Omega = B_R\subset \mathbb{R}^N$ with $N\geq 2$, and
let $u\in C^1 (\overline{B}_R)$ be a radially symmetric,
non-increasing function such that $u\geq 0$ and $u$ is a minimizer
of $I_{\lambda}$ with $I_{\lambda}(u) = m < 0$. Then, $u$ does
not vanish in $B_R$, that is, $u (x) > 0$
for all $x\in B_R$.
\end{lemma}
\begin{proof} Since $g$ is discontinuous at zero, we note that the
conclusion does not follow directly from uniqueness of solution
for the Cauchy problem with data at $r=R$ (In fact, writing
$u=u(r)$, $r=|x|$, we may have $u(R)=u'(R) = 0$ and
$u\not\equiv 0$).
Now, since $u\not\equiv 0$ by assumption,
$R_0 := \inf\{ r\leq R\ |\ u(s)=0 \mbox{ for $r\leq s\leq R$} \}$
satisfies
$0 < R_0 \leq R$. If $R_0 = R$ there is nothing to prove in view of the
fact that $u$ is non-increasing. On the other hand, if $R_0 < R$
then $u'(R_0)=0$ and $u(r) > 0$ for
$r\in [0, R_0)$. It is not hard to prove that this contradicts
that $u$ is a minimizer of $I_{\lambda}$. Indeed, if $R_0 < R$
then
$$
I_{\lambda}(u)=\frac{1}{2}\int_{B_{R_0}}|\nabla u|^2\,dx - \lambda
\int_{B_{R_0}}G(u)\,dx = m < 0\,.
$$
A simple calculation shows that the re-scaled function
$v(r)=u(\frac{R_0 r}{R})\in H^1_0 (B_R)\cap C^1(\overline{B_R})$
satisfies
$$
I_{\lambda}(v)= \delta^{2-N}\Big[
\frac{1}{2}\int_{B_{R_0}}|\nabla u|^2\,dx - \delta^{-2}\lambda
\int_{B_{R_0}}G(u)\,dx \Big],
$$
where $\delta:=R_0/R$ is less than $1$. Therefore, since we are
assuming $N\geq 2$, we would reach the contradiction $I_{\lambda}(v) < m$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm0.1}]
We observe that the functional $I_{\lambda}$ is weakly lower
semi-continuous on $H^1_0 (\Omega)$. Moreover, the sublinearity
assumption (F1) on $g(u)$
implies that $I_{\lambda}$ is coercive. Therefore, the infimum of
$I_{\lambda}$ is attained at some $\widehat{u}_{\lambda}$:
$$
\inf_{u\in H^1_0}I_{\lambda} (u) =
I_{\lambda}(\widehat{u}_{\lambda}).
$$
And, in view of Lemma \ref{lemma3}, $\widehat{u}_{\lambda}\in
C^{1,\epsilon}(\overline{\Omega})$ is a nonnegative solution of
\eqref{hPl}. We now claim that
$\widehat{u}_{\lambda}$ is nonzero for all $\lambda > 0$ large.
\smallskip
\noindent \textbf{Claim:} There exists $\Lambda > 0$ such that
$I_{\lambda}(\widehat{u}_{\lambda}) < 0$ for all $\lambda\geq \Lambda$.
In order to prove the claim it suffices to exhibit an element
$\widehat{w}\in H^1_0 (\Omega)$ such that $I_{\lambda}(\widehat{w}) < 0$ for all
$\lambda > 0$ large. This is quite standard considering that
$G(\delta) > 0$ by (F3). Indeed, letting
$\Omega_{\epsilon}:=\{x\in\Omega\ |\,dist(x,\partial\Omega) > \epsilon \}$
for $\epsilon> 0$ small, define $\widehat{w}$ so that $\widehat{w} (x) = \delta$ for
$x\in\Omega_{\epsilon}$ and $0\leq\widehat{w} (x)\leq \delta$ for
$x\in\Omega\backslash\Omega_{\epsilon}$. Then
\begin{align*}
I_{\lambda}(\widehat{w})
&= \frac{1}{2}\|\widehat{w} \|^2 - \lambda\Big(
\int_{\Omega_{\epsilon}}G(\widehat{w})\,dx +
\int_{\Omega\backslash\Omega_{\epsilon}}G(\widehat{w})\,dx \Big) \\
& \leq \frac{1}{2}\|\widehat{w} \|^2 - \lambda\big(
G(\delta)meas(\Omega_{\epsilon}) -
C(1+\delta^2)meas(\Omega\backslash\Omega_{\epsilon})\big)\,,
\end{align*}
where we note that the expression in the above parenthesis is
positive if we choose $\epsilon > 0$ sufficiently small.
Therefore, there exists $\Lambda > 0$ such that $I_{\lambda}(\widehat{w})
< 0$ for all $\lambda\geq\Lambda$, which proves the claim.
\smallskip
On the other hand, when $\Omega = B_R$, let
$\widehat{u}^{\ast}_{\lambda}$ denote the {\it Schwarz
Symmetrization} of $\widehat{u}_{\lambda}$, namely,
$\widehat{u}^{\ast}_{\lambda}$ is the unique radially symmetric,
non-increasing, nonnegative function in $H^1_0 (B_R)$ which is
equi-measurable with $\widehat{u}_{\lambda}$.
As is well known,
$$
\int_{B_R}G(\widehat{u}^{\ast}_{\lambda})\,dx =
\int_{B_R}G(\widehat{u}_{\lambda})\,dx
$$
and $\|\widehat{u}^{\ast}_{\lambda}\| \leq\| \widehat{u}_{\lambda}\|$, so
that $I_{\lambda}(\widehat{u}^{\ast}_{\lambda})\leq
I_{\lambda}(\widehat{u}_{\lambda})$. Therefore, we necessarily
have $I_{\lambda}(\widehat{u}^{\ast}_{\lambda}) =
I_{\lambda}(\widehat{u}_{\lambda})$ and may assume that
$\widehat{u}_{\lambda}=\widehat{u}^{\ast}_{\lambda}$. Moreover,
$\widehat{u}_{\lambda}> 0$ in $\Omega$ by Lemma \ref{lemma5}.
Therefore, $\widehat{u}_{\lambda}$ is a positive solution of both
\eqref{Pl} and \eqref{hPl}
Next, we recall that $u=0$ is a strict local minimum of
$I_{\lambda}$ by Lemma \ref{lemma1}. Therefore, since
$I_{\lambda}$ satisfies the Palais-Smale condition by Lemma
\ref{lemma4}, we can use the minima $u=0$ and
$u=\widehat{u}_{\lambda}$ of $I_{\lambda}$ to apply the Mountain
Pass Theorem and conclude that there exists a second nontrivial
critical point $\widehat{v}_{\lambda}$ with
$I_{\lambda}(\widehat{v}_{\lambda}) > 0$. Again,
$\widehat{v}_{\lambda}$ is a nonnegative solution of
\eqref{hPl} in view of Lemma \ref{lemma3}. In
addition, when $\Omega = B_R$, arguments similar to those in
\cite[Theorem 3.4]{bonaBT} (see pp. 403-405) show that we may
assume $\widehat{v}_{\lambda}=\widehat{v}^{\ast}_{\lambda}$. The
proof of Theorem \ref{thm0.1} is complete.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm0.2}]
As is well-known, the
superlinearity condition (F2) readily implies the existence of
an element $e_{\lambda}\in H^1_0 (\Omega)$ such that
$I_{\lambda}(e_{\lambda}) \leq 0$. In fact, the weaker condition
$\lim_{s\to +\infty}F(s)/s^2=+\infty$ suffices for that
purpose. On the other hand, Lemma \ref{lemma2} says that $u=0$ is
a (strict) local minimum of $I_{\lambda}$ and Lemma \ref{lemma4}
says that $I_{\lambda}$ satisfies $(PS)_c$ for every
$c\in\mathbb{R}$. Therefore, an application of the Mountain-Pass
Theorem stated in section 2 yields the existence of a critical
point $\widehat{v}_{\lambda}$ such that
$$
I_{\lambda}(\widehat{v}_{\lambda}) > 0.
$$
In particular, $\widehat{v}_{\lambda}\neq 0$, and it follows that
$\widehat{v}_{\lambda}$ is a nonnegative solution of
\eqref{hPl} by Lemma \ref{lemma3}. As in the proof
of Theorem \ref{thm0.1}, we may assume that
$\widehat{v}_{\lambda}=\widehat{v}^{\ast}_{\lambda}$ in the case
of a ball $\Omega = B_R$.
Finally, still in the case of a ball $\Omega = B_R$, we claim that
there exists $\Lambda_1 > 0$ such that, for all $0 < \lambda <
\Lambda_1$, $\widehat{v}_{\lambda} = \widehat{v}^{\ast}_{\lambda}$
is a positive solution of \eqref{hPl} (hence
of \eqref{Pl}) having negative normal derivative on
$\partial B_R$.
Indeed, if that is not the case then, for any given $\lambda > 0$,
we can find $0 < \mu =\mu(\lambda) < \lambda$ such that the
nonnegative solution $\widehat{v}_{\mu}=\widehat{v}^{\ast}_{\mu}$
of $(\widehat{P}_{\mu})$ obtained above satisfies
$$
\widehat{v}_{\mu}(r) > 0\quad \mbox{for }r\in [0,R_0 ),\quad
\widehat{v}_{\mu}'(R_0)=0 \quad \mbox{and}\quad
\widehat{v}_{\mu}(r)=0\quad\mbox{for }r\in [R_0 ,R],
$$
for some $0 < R_0 \leq R$. Therefore, the re-scaled function
$\widehat{w}_{\mu}(r):=\widehat{v}_{\mu}(\frac{R_0 r}{R})$ is a
positive solution of $(P_{\mu_0})$ (again in the ball
$B_R$), with $\mu_0 := \mu R_0^2 /R^2 \leq \mu$. This shows that
we can always construct a decreasing sequence $\mu_n > 0$
satisfying alternative $(ii)$ of Theorem \ref{thm0.2}, in case alternative
$(i)$ does not hold.
\end{proof}
\section{Final Remarks}
As we shall explain, the results in both Theorem \ref{thm0.1} and Theorem
\ref{thm0.2} concerning the semipositone problem \eqref{Pl} in a ball
are optimal in a sense to be made clear in what follows.
\subsection{The Sublinear Case}
In view of the paper \cite{castroHS} we know that, in case
$\Omega$ is a bounded domain with a convex outer boundary, problem
\eqref{Pl} has a unique nonnegative solution for all $\lambda
> 0$ large provided that, in addition to (F0), one assumes
\begin{itemize}
\item[(i)] $\lim_{s\to\infty}f(s)=\infty$,
\item[(ii)] $\lim_{s\to\infty} f(s)/s=0$,
\item[(iii)] $f$ is increasing and concave.
\end{itemize}
Furthermore, it is shown in \cite{castroHS} that this unique
nonnegative solution is in fact positive in $\Omega$.
Therefore, under these hypotheses, we conclude that at least one
of the two solutions obtained in Theorem \ref{thm0.1} has to have a large
zero-set in the sense that
$\mathop{\rm meas} \{ x\in\Omega\ |\ u(x)=0\}> 0$
(since, as we mentioned in the Introduction, a nontrivial
solution of \eqref{hPl} with $meas \{ x\in\Omega\ |\
u(x)=0\} = 0$ is a nonnegative solution of \eqref{Pl}).
Moreover, in the specific case of a ball $\Omega = B_R$, we know
from Theorem \ref{thm0.1} that both nonnegative solutions of
\eqref{hPl} are radially symmetric and
non-increasing, with one of them, say $\widehat{u}_{\lambda}$,
being in fact the unique positive solution of \eqref{Pl} for
$\lambda > 0$ large. Therefore, the second solution
$\widehat{v}_{\lambda}$ must necessarily satisfy
$\widehat{v}_{\lambda}(r) > 0$ for $0\leq r< R_0$,
$\widehat{v}_{\lambda}(r) = 0$ for $R_0\leq r\leq R$, and
$\widehat{v}_{\lambda}'(R_0) = 0$, for some $0 < R_0 < R$
(recall that, by Lemma 2.4, we have
$\widehat{v}_{\lambda}\in C^{1,\epsilon}(\overline{\Omega})$).
Therefore, the natural
extension of $\widehat{v}_{\lambda}$ to $\mathbb{R}^N$, by letting
$\widehat{v}_{\lambda}=0$ outside $B_R$, yields a {\it bump
(compactly supported) solution} of
$$
- \Delta u = \lambda g(u)\quad \mbox{in }\mathbb{R}^N.
$$
\subsection{The Superlinear Case}
In view of the paper \cite{brownCS}, it is known that when
$\Omega$ is a ball $B_R$, problem \eqref{Pl} has no
nonnegative radially symmetric solution for all $\lambda > 0$
sufficiently large provided that, in addition to (F0), one
assumes
\begin{itemize}
\item[(i)] $\liminf_{s\to\infty}\frac{f(s)}{s^{\alpha}} > 0$
(for some $\alpha > 1$),
\item[(ii)] $f$ is increasing.
\end{itemize}
Therefore, for $\lambda > 0$ large, the nonnegative solution
$\widehat{v}_{\lambda}$ obtained in Theorem \ref{thm0.2} for the case of
the ball $B_R$ must have a {\it large} zero-set. It follows,
similarly to the previous case, that the natural extension of
$\widehat{v}_{\lambda}$ to $\mathbb{R}^N$ yields again a {\it bump
solution} of
$$
- \Delta u = \lambda g(u)\quad \mbox{in }\mathbb{R}^N.
$$
Secondly, a result in \cite{castroS} yields existence of a
positive radially symmetric solution for \eqref{Pl} when
$\lambda > 0$ is small and, in addition to (F0), one
assumes suitable technical conditions on the superlinearity $f$.
Moreover, such a solution is shown to have negative normal
derivative on $\partial B_R$. We thus see that, for appropriate
classes of $f$'s, alternative $(i)$ of Theorem \ref{thm0.2} must hold true.
On the other hand, under still further conditions on $f$, it is
shown in \cite{aliCS} that the above positive solution obtained in
\cite{castroS} is unique, thus precluding alternative (ii)
of Theorem \ref{thm0.2}. It would be interesting to find out whether
alternative (ii) can indeed occur in some other superlinear
situations.
\subsection*{Acknowledgements}
This work was initiated when
the first author was visiting the Wuhan Institute of Physics and
Mathematics of the Chinese Academy of Sciences. He thankfully
acknowledges their kind hospitality.
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\end{document}