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\title[Lane--Emden equations]
{Multiple solutions for Lane--Emden equations with mixed nonlinearities}

\author{Vicen\c{t}iu D. R\u{a}dulescu}



\address{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, 014700 Bucharest, Romania \& Department of Mathematics, University of Craiova,  200585 Craiova, Romania.} \email{vicentiu.radulescu@@imar.ro}

\keywords{Lane--Emden equation, eigenvalue problem, multiple solutions.}
\thanks{{\em 2010 Mathematics Subject Classification.} 35J60, 58J05.}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{Definition}{Definition}[section]
\newtheorem{example}{Example}[section]
\newtheorem{th}{Theorem}
\renewcommand{\theth}{\Alph{th}}
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\def\essinf{\mathop {\rm ess\,inf}}
\def\esssup{\mathop {\rm ess\,sup}}
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\def\essinf{\mathop {\rm ess\,inf}}
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\def\ri{\rightarrow}
\def\intom{\int_\Omega}
\def\huo{H^1_0(\Omega)}
\begin{document}


%%% ----------------------------------------------------------------------

\begin{abstract}
We are concerned with a linear perturbation of the Lane-Emden equation with different growths near the origin and at
infinity. By means of a version of the Pucci--Serrin three critical points theorem, we establish the existence of at least two nontrivial solutions in the case of large values of the parameter.
\end{abstract}
%%% ----------------------------------------------------------------------
\maketitle

\bigskip
\begin{center}{\small\it This paper is dedicated with esteem to Professor Jean Mawhin on his 70th birthday anniversary} \end{center}

\bigskip
The Lane--Emden equation describes naturally many physical phenomena.
For example, super-diffusivity equations of this type have been proposed
by de Gennes \cite{gennes} as a model for long-range Van der Waals interactions in thin
films spreading on solid surfaces. This equation also appears in the study of cellular
automata and interacting particle systems with self-organized criticality (see \cite{chayes}),
as well as to describe the flow over an impermeable plate \cite{calle}. Our main purpose in the present paper is to connect a general class of Lane--Emden equations with the eigenvalue problem for the Laplace operator in order to establish a striking multiplicity result
for large values of a certain real parameter. The proof of this existence property for the perturbed equation relies
on simple variational tools, namely on a version of the celebrated Pucci--Serrin three critical points theorem (see \cite{pucser}).

\section{Preliminary results}
Let $\Omega\subset\R^N$ be an open bounded set with smooth boundary. We first consider the Lane--Emden problem
\begin{equation}\label{le}
 \left\{\begin{tabular}{lll}
&$-\Delta u=|u|^{p-1}u$& \qquad  $\mbox{in}\
\Omega$\\
&$u=0$ &\qquad  $\mbox{on}\
\partial\Omega$\\
&$u\not\equiv0$ &\qquad  $\mbox{in}\
\Omega$,
\end{tabular} \right.
\end{equation}
where $p$ is a positive number. If $0<p<1$ then, by the Brezis-Oswald theorem \cite{1}, problem \eqref{le} has a unique positive solution. If $p=1$, the existence of a solution depends on the spectrum of the Laplace operator in $\huo$. Next, if $1<p<(N+2)/(N-2)$ (or if $1<p<\infty$ if $N\in\{1,2\}$) then problem \eqref{le} has a solution. This is a direct consequence of the mountain pass theorem of Ambrosetti and Rabinowitz \cite{ar1}.

Existence or non-existence results have been estabished both in the critical and in the supercritical case according to various assumptions on $\Omega$. For instance, if $\Omega$ is starshaped, a direct application of the Pohozaev identity shows that problem \eqref{le} does not have any solution in the supercritical case. Related results have been established for various perturbations of problem \eqref{le}. Indeed, let us consider the problem
\begin{equation}\label{le1}
 \left\{\begin{tabular}{lll}
&$-\Delta u=|u|^{p-1}u+\lambda u$& \qquad  $\mbox{in}\
\Omega$\\
&$u=0$ &\qquad  $\mbox{on}\
\partial\Omega$\\
&$u\not\equiv0$ &\qquad  $\mbox{in}\
\Omega$,
\end{tabular} \right.
\end{equation}
with $1<p<(N+2)/(N-2)$. Let $\lambda_1$ denote the first eigenvalue of $(-\Delta)$. Then, by the mountain pass theorem, problem \eqref{le1} has a {\it positive} solution provided that $\lambda<\lambda_1$. If $\lambda\geq\lambda_1$ then problem \eqref{le1} does not have any positive solution, as we observe easily by multiplying with $\phi_1$ and integrating (as usual, $\phi_1$ denotes the first eigenfunction of the Laplace operator). A refined result, whose proof relies on the dual variational method, establishes that for {\it all} $\lambda\geq\lambda_1$, problem \eqref{le1} has at least one solution.

The second major issue in this paper is the eigenvalue problem associated to the Laplace operator. As stated in Zworski \cite{zwo}, ``eigenvalues describe, among other things,
the energies of bound states, states that exist forever if unperturbed. These do exist in real
life [...]. In most situation however, states do not exist for ever, and a more accurate model is
given by a decaying state that oscillates at some rate. Eigenvalues are yet another expression
of humanity's narcissist desire for immortality."

Consider the weighted eigenvalue problem
the eigenvalue problem
\begin{equation}\label{eig}
 \left\{\begin{tabular}{lll}
&$-\Delta u=\lambda a(x)u$& \qquad  $\mbox{in}\
\Omega$\\
&$u=0$ &\qquad  $\mbox{on}\
\partial\Omega$\\
&$u\not\equiv0$ &\qquad  $\mbox{in}\
\Omega$.
\end{tabular} \right.
\end{equation}
Problems of this type have a long history. If $\Omega$ is bounded
and $a\equiv 1$, problem \eqref{eig} is related to the Riesz-Fredholm theory
of self-adjoint and compact operators (see, Theorem VI.11 in Brezis \cite{Brezis}). The case of a %%@
non-constant
potential $a$ has been first considered in the pioneering papers
of Bocher \cite{bo}, Hess and Kato \cite{hk}, Minakshisundaran and  Pleijel \cite{mp} and Pleijel %%@
\cite{p}.
For instance, Minakshisundaran and  Pleijel \cite{mp}, \cite{p} studied
the case where $\Omega$ is bounded, $a\in L^\infty (\Omega)$,
$a\geq 0$ in $\Omega$
and $a>0$ in $\Omega_0\subset\Omega$ with $|\Omega_0|>0$.

\section{Main result}
This paper is strongly inspired by the celebrated work by Brezis and Oswald \cite{1}, which is concerned with the qualitative analysis of solutions of sublinear elliptic equations with Dirichlet boundary condition. By contrast, in the present paper we deal with nonlinearities having a mixed behaviour. More precisely, we study the nonlinear problem
\begin{equation}\label{equ}
 \left\{\begin{tabular}{lll}
&$-\Delta u= a(x)u+\lambda f(u)$& \qquad  $\mbox{in}\
\Omega$\\
&$u=0$ &\qquad  $\mbox{on}\
\partial\Omega$\\
&$u\not\equiv0$ &\qquad  $\mbox{in}\
\Omega$,
\end{tabular} \right.
\end{equation}
where $\lambda$ is a positive parameter, $a\in L^\infty(\Omega)$, and $a>0$ in $\Omega_0\subset\Omega$ with $|\Omega_0|>0$. We assume that $f:\R\ri\R$ is a continuous function such that
\begin{equation}\label{hyp2}
\mbox{there exists $z>0$ such that $f\geq 0$ and $f\not\equiv 0$ in $[0,z]$}\end{equation}
and, for some $0<q<1<p$,
\begin{equation}\label{hyp1}
\sup_{t\in\R}\frac{|f(t)|}{|t|^p}<\infty\qquad\mbox{and}\qquad \sup_{t\in\R}\frac{|f(t)|}{1+|t|^q}<\infty\,.\end{equation}

In particular, the function
$$f(t)=\left\{\begin{array}{lll}
& |t|^{p-1}t&\qquad\mbox{if $|t|\leq 1$}\\
& |t|^{q-1}t&\qquad\mbox{if $| t|> 1$}\end{array}\right.$$
fulfills assumptions \eqref{hyp2} and \eqref{hyp1}.

Let $F(t):=\int_0^tf(s)ds$. Then assumption \eqref{hyp1} implies that there is $C>0$ such that for all $u\in\R$,
 \begin{equation}\label{hyp3}
|F(u)|\leq C|u|^{p+1}\qquad\mbox{and}\qquad |F(u)|\leq C(1+|u|^{q+1})\,.\end{equation}

Let $\lambda_1>0$ denote the first eigenvalue of the weighted eigenvalue problem \eqref{eig}.

Our main result is the following multiplicity property.

\begin{theorem}\label{teo1}
Let $\Omega$ be a bounded domain with smooth boundary and assume that $f$ satisfies hypotheses \eqref{hyp2} and \eqref{hyp1}. Assume that $\lambda_1>1$. The there exists $\lambda^*\geq 0$ such that problem \eqref{equ} has at least two solutions, provided that $\lambda >\lambda^*$.
\end{theorem}

{\it Proof of Theorem \ref{teo1}}. The weak solutions of problem \eqref{equ} are the critical points of the associated energy functional ${\mathcal E}:\huo\ri\R$ defined by ${\mathcal E}={\mathcal E}_1+\lambda{\mathcal E}_2,$ where
$${\mathcal E}_1(u)=\frac 12\intom (|\nabla u|^2-a(x)u^2)dx\qquad\mbox{and}\qquad {\mathcal E}_2(u)=-\intom F(u)dx.$$

We split the proof into several parts.

\smallskip
{\sc Claim 1.} {\it The mapping $\huo\ni u\longmapsto F(u)$ is not constant.}

Let $\varphi_1>0$ be the first eigenfunction of problem \eqref{eig}. Returning to hypothesis \eqref{hyp2}, we set
$\omega:=\{x\in\Omega;\ \varphi_1(x)>z\}$. Then $\omega$ is open and nonempty (this follows after replacing eventually $\varphi_1$ with $t\varphi_1$ for $t$ large enough). Set $v:=\psi\circ\varphi_1$, where $\psi (t):=\min\{t,z\}$. Then $v\in\huo$, $0\leq v\leq z$ in $\Omega$, and $v\equiv z$ in $\omega$. Therefore $F(v)\geq 0$ in $\Omega$ and $F(v(x))>0$ for all $x\in\omega$. This concludes the proof of Claim 1.

\smallskip
{\sc Claim 2.} {\it ${\mathcal E}$ satisfies the Palais-Smale condition.}

We first observe that ${\mathcal E}_1$ is coercive. Indeed, using the variational characterization of $\lambda_1$, we deduce that for all $u\in\huo$,
$${\mathcal E}_1(u)=\frac 12\,\left(1-\frac{1}{\lambda_1}\right)\intom|\nabla u|^2 dx=\alpha\,\| u\|^2\,,$$
where $\alpha>0$.

Next, using assumption \eqref{hyp1} and Sobolev embeddings, we obtain
$${\mathcal E}_2(u)\geq -C\lambda\intom |u|^{p+1}dx\geq -C\lambda\, \|u\|^{q+1}.$$
Since $q\in (0,1)$ we conclude that ${\mathcal E}$ is coercive. Thus, by Lemma~V.4 in Brezis and Nirenberg \cite{brenirbook}, we conclude that ${\mathcal E}$ satisfies the Palais-Smale condition.

\smallskip
{\sc Step 3.} {\it Proof of Theorem \ref{teo1} concluded.}

The classical three critical points theorem of Pucci and Serrin \cite{pucser} (see also \cite[Theorem 1.14]{KRV}) establishes that if $X$ is a Banach space and $J:X\ri\RR$ is of class $C^1$ and satisfies the Palais-Smale condition and has two local minima, then $J$ has at least three distinct critical points. Thus, to conclude the proof, it is enough to argue that ${\mathcal E}$ has at least two local minima provided that $\lambda$ is large enough. For such a purpose we consider the functionals
$$\Phi(t):=\inf_{{\mathcal E}_2(u)<t}\frac{\inf_{{\mathcal E}_2(v)=t}{\mathcal E}_1(v)-{\mathcal E}_1(u)}{{\mathcal E}_2(u)-t}\,;\qquad t\in\R$$
and
$$\Psi (t):=\sup_{{\mathcal E}_2(u)>t}\frac{\inf_{{\mathcal E}_2(v)=t}{\mathcal E}_1(v)-{\mathcal E}_1(u)}{{\mathcal E}_2(u)-t}\,;\qquad t\in\R\,.$$

A straightforward computation shows that
$$\limsup_{t\ri 0^-}\Phi(t)\leq\Phi(0)\qquad\mbox{and}\qquad\liminf_{t\ri 0^-}\Psi(t)=+\infty\,.$$
Thus, if $\lambda>\lambda^*:=\Phi(0)$, then ${\mathcal E}$ has at least two distinct nontrivial critical points. This concludes the proof. \qed

\medskip
 \indent {\bf Acknowledgements.}  The author acknowledges the support through  Grant CNCS PCE--47/2011.

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\end{document} 