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\begin{document}

\begin{frontmatter}


\selectlanguage{english}
\title{Low and high energy solutions of nonlinear elliptic\\ oscillatory problems}

\vspace{-2.6cm}

\selectlanguage{french}
\title{Solutions \`a basse et haute \'energie pour des probl\`emes elliptiques non lin\'eaires avec terme oscillatoire}

\selectlanguage{english}
\author[GM]{Giovanni Molica Bisci}
\ead{gmolica@unirc.it}
\author[VR1,VR2]{Vicen\c tiu R\u{a}dulescu}
%\thanks[label3]{Correspondence address: Vicen\c tiu R\u{a}dulescu, University of Craiova, Department of Mathematics, 200585 Craiova, Romania. E-mail: {\tt
%vicentiu.radulescu@imar.ro}}
\ead{vicentiu.radulescu@imar.ro}
%\ead[url]{http://www.math.ucv.ro/\~{}radulescu}
\author[RS]{Raffaella Servadei}
\ead{servadei@mat.unical.it}
\address[GM]{Dipartimento Patrimonio, Architettura e Urbanistica, University of Reggio Calabria,\\ Feo di Vito, 89124 Reggio Calabria, Italy}
\address[VR1]{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,  014700
Bucharest, Romania}
\address[VR2]{Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585 Craiova, Romania}
\address[RS]{Dipartimento di Matematica e Informatica,
          Universit\`a della Calabria,
          Ponte Pietro Bucci 31 B,\\
           87036 Arcavacata di Rende (Cosenza), Italy}

\begin{abstract}
% Text of abstract in English
In this Note, we study the existence of low or high energy solutions for a class of elliptic problems containing a nonlinear term that oscillates either near the origin or at infinity. We point out the competition effect between the oscillatory nonlinearity, a polynomial growth term, and the values of a real parameter.  The proofs combine related variational methods.
\vskip 0.5\baselineskip

\selectlanguage{francais}
% Text of abstract in French
\noindent{\bf R\'esum\'e} \vskip 0.5\baselineskip \noindent
Dans cette Note, nous  \'etudions l'existence de solutions \`a basse ou \`a haute \'energie pour une classe de probl\`emes elliptiques contenant un terme non lin\'eaire oscillatoire autour de l'origine ou \`a l'infini. Nous mettons en \'evidence l'effet de competition entre la nonlin\'earit\'e oscillatoire, le terme \`a croissance polynomiale et les valeurs d'un param\`etre r\'eel. Les preuves combinent des m\'ethodes topologiques et variationnelles.
\end{abstract}

\end{frontmatter}

% now the Version fran\c{c}aise abr\'eg\'ee, if it exists
\selectlanguage{francais}
\section*{Version fran\c{c}aise abr\'eg\'ee}
Soit $\Omega\subset \RR^N$ ($N\geq 3$) un domaine born\'e et r\'egulier, $\beta\in L^\infty(\Omega)$,  $\lambda\in\RR$, $q>0$ et $f:[0,\infty)\ri\RR$ une fonction continue qui oscille autour de l'origine ou \`a l'infini. Nous supposons que $A:\overline\Omega\times\RR^N\ri\RR^N$ est une fonction continue telle
que pour tout $(x,\xi)\in\Omega\times\RR^N$
$$A(x,\xi)\cdot\xi\geq\Gamma_1|\xi|^p\quad\mbox{et}\quad |A(x,\xi)|\leq\Gamma_2|\xi|^{p-1},$$
o\`u $p>1$ et $\Gamma_1,\, \Gamma_2>0$.

Dans cette Note nous \'etudions
le probl\`eme non lin\'eaire
$$
\left\{
\begin{array}{ll}
-{\rm div}\, A(x,\nabla u)= \lambda \beta(x)u^q+f(u) &\rm dans \ \Omega
\\
u\geq 0&\rm dans\ \Omega\\
 u=0&\rm sur \
\partial \Omega.
\end{array}
\right. \eqno(P)
$$

Le premier r\'esultat de cette Note porte sur le cas o\`u $f$ a des oscillations autour de l'origine. Nous montrons d'abord que le probl\`eme $(P)$ a une infinit\'e de solutions ``\`a basse \'energie"  si $q\geq p-1$ et au moins un nombre fini de solutions si $0<q<p-1$. Plus pr\'ecis\'ement, si $q\geq p-1$, nous montrons l'existence d'une suite $\{u_j\}\subset W_0^{1,p}(\Omega)$ de solutions faibles du probl\`eme $(P)$ telle que
$${\displaystyle \lim_{j\ri +\infty}\|u_j\|_{W_0^{1,p}(\Omega)}=\lim_{j\ri +\infty}\|u_j\|_{L^\infty(\Omega)}=0.}$$


Dans le cas o\`u $f$ a des oscillations \`a l'infini, il existe une infinit\'e de solutions $\{u_j\}\subset W_0^{1,p}(\Omega)$  si $0<q\leq p-1$ et au moins un nombre fini de solutions si $q>p-1$. De plus, si $0<q\leq p-1$, alors ${\displaystyle \lim_{j\ri +\infty}\|u_j\|_{L^\infty(\Omega)}=+\infty.}$




\vspace{0.1cm}
$$\line(1,0){300}$$
\selectlanguage{english}

\vspace{0.2cm}
Competition phenomena in elliptic equations have been widely studied in the literature in different contexts. After the seminal work~\cite{ABC}, where Ambrosetti, Brezis and Cerami studied a Laplacian equation involving a concave-convex nonlinearity, a lot of papers appeared on this subject.
Also when dealing with singular terms, the interactions with different type of nonlinearities were investigated: see, for instance, Ghoussoub and Yuan \cite{ghoussoub}, Pucci and Servadei \cite{PS1}, \cite{PS2} for equations involving superlinear and subcritical terms.

In this Note, we are interested in problems driven by general operators of $p$-Laplacian type involving oscillatory terms, in presence of a concave or convex power. Usually,  equations involving oscillatory nonlinearities give infinitely many distinct solutions, but the presence of an additional term may alter the situation.

 Let $\Omega\subset \RR^N$ ($N \geq 3$) be a bounded domain with smooth boundary, $q>0$, $\lambda\in \RR$,  and let  $f:[0,+\infty)\to \RR$ be a continuous function. Suppose that $\beta \in L^{\infty}(\Omega)$ is a potential that is {\it indefinite} in sign. We also assume that $A:\overline \Omega \times \RR^N\to \RR^N$ is a continuous function such that
$$
 A(x,\xi)\cdot \xi\geq \Gamma_1 |\xi|^p\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,|A(x,\xi)|\leq \Gamma_2|\xi|^{p-1}\,\,\, \mbox{for all}\, (x,\xi)\in \Omega\times\RR^N\,,
$$
for some $p>1$ and $0<\Gamma_1\leq \Gamma_2$.
Suppose that $A$ derives from a potential, namely
$A=\nabla_\xi a$,
where $a:\overline\Omega\times \RR^N\to \RR$ is continuous,
$a(x,0)=0$, $a(x,\xi)=a(x,-\xi)$ for all $(x, \xi)\in\Omega\times \RR^N$, and
$a(x,\cdot)$ is strictly convex in $\RR^N$ for all $x\in \Omega$.

We are concerned with the nonlinear Dirichlet problem
\begin{equation}\label{problem}
\left\{ \begin{array}{ll}
-{\rm div}A(x,\nabla u) = \lambda \beta(x) u^q + f(u) & \mbox{in} \, \Omega\\ u\geq 0 & \mbox{in} \, \Omega\\ u=0 & \mbox{on} \, \partial \Omega. \end{array} \right.
\end{equation}

\section{Oscillation near the origin}
Set $F(s):=\displaystyle\int_0^sf(t)dt$ and assume that
\begin{equation}\label{f2origin}
\displaystyle \liminf_{s\to 0^+} \frac{f(s)}{s^{p-1}}=:-\ell_0\in [-\infty, 0)\qquad-\infty < \liminf_{s\to 0^+} \frac{F(s)}{s^p}\leq \limsup_{s\to 0^+} \frac{F(s)}{s^p} = +\infty\,.
\end{equation}
%\begin{equation}\label{f1origin}
%\displaystyle -\infty < \liminf_{s\to 0^+} \frac{F(s)}{s^p}\leq \limsup_{s\to 0^+} \frac{F(s)}{s^p} = +\infty\,.
%\end{equation}

{\bf Examples.} (i) Assume that  $\alpha$, $\sigma$, $\gamma\in\RR$ satisfy $1<\sigma+1<\alpha<p$ and $\gamma>0$. Define
\begin{equation}\label{exa1}f(s)=\left\{\begin{array}{ll}
\alpha s^{\alpha-1}\Big(1-\sin s^{-\sigma}\Big)+\sigma s^{\alpha-\sigma-1} \cos s^{-\sigma}-p\gamma s^{p-1} & \mbox{if}\,\,\, s>0\\
0 & \mbox{if}\,\,\, s=0\,,
\end{array}\right.\end{equation}

(ii) Assume that $\alpha$, $\sigma$ and $\gamma\in\RR$ are such that $1<\alpha<p$, $\sigma>0$, $\alpha-\sigma>1$  and $\gamma>0$. Define
\begin{equation}\label{exa2}f(s)=\left\{\begin{array}{ll}
\alpha s^{\alpha-1}\cos^2 s^{-\sigma}-2\sigma s^{\alpha-\sigma-1} \cos s^{-\sigma}\sin s^{-\sigma}-p\gamma s^{p-1} & \mbox{if}\,\,\, s>0\\
0 & \mbox{if}\,\,\, s=0\,,
\end{array}\right.\end{equation}

Then the functions defined by relations (\ref{exa1}) and (\ref{exa2}) have oscillation near the origin, in the sense described by hypothesis (\ref{f2origin}).

The main result in this section is the following.
\begin{theorem}\label{thorigin}
Assume that $f$ satisfies condition $($\ref{f2origin}$)$. If either
\begin{enumerate}
\item[$ a)$] $q = p-1$, $\ell_0\in (0, +\infty)$ and $\lambda \beta(x) < \lambda_0$ a.e. $x\in \Omega$ for some $\lambda_0\in (0, \ell_0)$ or
\item[$ b)$] $q = p-1$, $\ell_0=+\infty$ and $\lambda \in \RR$ is arbitrary or
\item[$ c)$] $q >p-1$ and $\lambda \in \RR$ is arbitrary,
\end{enumerate}
then there exists a sequence $\{u_j\}_j$ in $W_0^{1,p}(\Omega)$ of distinct weak solutions of problem$~($\ref{problem}$)$ such that
$${\displaystyle\lim_{j\to +\infty} \|u_j\|_{W^{1,p}_0(\Omega)}
= \lim_{j\to +\infty} \|u_j\|_{L^{\infty}(\Omega)}=0.}$$

Assume that $0<q<p-1$. Then for every $k\in \NN$ there exists $\Lambda_k>0$ such that problem$~($\ref{problem}$)$ has at least $k$ distinct weak solutions $u_1, \dots, u_k\in W^{1,p}_0(\Omega)$
such that
$\|u_j\|_{W^{1,p}_0(\Omega)}\leq 1/j$ and $\|u_j\|_{L^{\infty}(\Omega)}\leq 1/j$ for all $j=1,\dots, k$,
provided that $|\lambda|<\Lambda_k$.
\end{theorem}
\smallskip
{\it Sketch of the proof.} Consider the auxiliary problem
\begin{equation}\label{auxiliary}
%\tag{$P_h^K$}
\left\{ \begin{array}{ll} -{\rm div}A(x,\nabla u) + K(x) |u|^{p-2}u = h(x, u)\,\,\,\, & \mbox{in} \, \Omega\\
 u=0 & \mbox{on} \, \partial \Omega.
 \end{array} \right.
\end{equation}
Throughout this Note we assume that $K \in L^{\infty}(\Omega)$ with $\displaystyle\mathop {\rm ess\,inf}_{x\in \Omega} K(x)>0$,
while $h:\Omega \times [0,+\infty)\to \RR$ is a Carath\'eodory function satisfying
$h(x, 0)=0$ for a.e. $x \in \Omega$.
Set $H(x,s):=\displaystyle\int_0^sh(x,t)dt,$
for all $s\in\RR$.

A key ingredient in the proof of Theorem \ref{thorigin} if $q\geq p-1$ is the following multiplicity property.
\smallskip
\begin{lemma}\label{thauxorigin}
Assume that the following hypotheses are fulfilled$:$
\begin{equation}\label{h1origin}
\mbox{there exists}\,\,\, \bar s > 0\,\,\, \mbox{such that}\\
\sup_{s\in [0,\bar s]} |h(\cdot, s)|\in L^{\infty}(\Omega)\,;
\end{equation}
\begin{equation}\label{h3origin}
\begin{array}{ll}
& \di\qquad  \mbox{there exist two sequences}\,\,\, \{\delta_j\}_j\,\,\mbox{and}\,\, \{\eta_j\}_j\,\,\, \mbox{with}\,\,\, 0 < \eta_{j+1} < \delta_j < \eta_j\,\, \mbox{and}\\
& \displaystyle\lim_{j\to +\infty} \eta_j = 0\,\, \mbox{such that}\,\,\, h(x, s) \leq 0\,\,\, \mbox{for a.e.}\, x\in \Omega\,\,\, \mbox{and for every}\, s\in [\delta_j,\eta_j],\,\, j\in \NN\,;
\end{array}
\end{equation}
\begin{equation}\label{h2origin}
-\infty< \displaystyle\liminf_{s\to 0^+}\displaystyle \frac{H(x,s)}{s^p}\leq\displaystyle\limsup_{s\to 0^+} \displaystyle\frac{H(x,s)}{s^p}=+\infty\,\, \mbox{uniformly for a.e.}\,\,\, x\in \Omega\,.
\end{equation}

Then there exists a sequence $\{u_j\}_j \subset W_0^{1,p}(\Omega)$ of distinct non-trivial non-negative weak solutions of problem~$($\ref{auxiliary}$)$ such that
${\displaystyle \lim_{j\to +\infty} \|u_j\|_{W^{1,p}_0(\Omega)}= \lim_{j\to +\infty} \|u_j\|_{L^{\infty}(\Omega)} = 0}$.
\end{lemma}
\smallskip

Returning to the proof of Theorem \ref{thorigin}, let us first assume that $q = p-1$, $\ell_0\in (0, +\infty)$, and $\lambda \in \RR$ is such that $\lambda \beta(x) < \lambda_0$ a.e. $x\in \Omega$ for some $\lambda_0\in(0,\ell_0)$. Let us choose $\tilde{\lambda}_0 \in (\lambda_0, \ell_0)$ and let
$
K(x) := \tilde{\lambda}_0 - \lambda \beta(x)$ and $h(x, s) := \tilde{\lambda}_0 s^{p-1} + f (s)$.

Next, we assume that  $q=p-1$, $\ell_0=+\infty$. and $\lambda \in \RR$. In this case we choose $\tilde{\lambda}_0 \in (\lambda_0, \ell_0)$ and
set
$
K(x) := \tilde{\lambda}_0$ and $h(x, s) := \Big(\lambda \beta(x)+\tilde{\lambda}_0\Big) s^{p-1} + f (s).
$

If $q>p-1$ and $\lambda \in \RR$, we take $\tilde\lambda_0\in (0,\ell_0)$ and define
$
K(x) := \tilde\lambda_0$ and $ h(x, s) := \lambda \beta(x) s^q + \tilde\lambda_0 s^{p-1} + f (s).$

In all these cases, by straightforward computation, we deduce that
$K$ and $h$  satisfy the assumptions of Lemma~\ref{thauxorigin}. Thus, problem~(\ref{auxiliary}) has infinitely many solutions $\{u_j\}_j$  satisfying ${\displaystyle \lim_{j\to +\infty} \|u_j\|_{W^{1,p}_0(\Omega)}}$
$= {\displaystyle \lim_{j\to +\infty} \|u_j\|_{L^{\infty}(\Omega)}=0}$.
Due to the choice of $K$ and $h$, we also obtain that $u_j$ is a weak solution of problem~(\ref{problem}).

Let us now assume that $0<q<p-1$. We associate to problem~(\ref{auxiliary}) the energy functional $\mathcal{E}_{K,\, h}:W_0^{1,p}(\Omega)\to \RR$  defined by
$
\mathcal{E}_{K,\, h}(u) = \int_\Omega a(x,\nabla u(x))\,dx+\frac{1}{p}\int_\Omega K(x)|u(x)|^p\,dx - \int_\Omega H(x,u(x))\,dx\,.
$

 The key ingredient in this case is the following result.
\smallskip
\begin{lemma}\label{thauxiliary}
Assume that the following hypotheses are fulfilled$:$
\begin{equation}\label{condhbis}
\mbox{there exists}\,\,\, M > 0\,\,\, \mbox{such that}\,\,\, |h(x, s)|\leq M\,\,\, \mbox{for a.e.}\, x\in \Omega\,\,\, \mbox{and for any}\, s \geq 0\,;
\end{equation}
\begin{equation}\label{hnegativo}
\begin{array}{ll}
& \mbox{there exist}\,\,\delta\, \mbox{and}\,\, \eta,\, \mbox{with}\,\, 0<\delta<\eta,\,\, \mbox{such that}\\
& h(x, s)\leq 0\,\,\, \mbox{for a.e.}\, x \in \Omega \,\,\, \mbox{and for any}\, s \in [\delta, \eta]\,.
\end{array}
\end{equation}

 Then
\begin{enumerate}
\item[$i)$] the functional {$\mathcal{E}_{K,\, h}$} is bounded from below on $W^{\eta}$ and its infimum is attained at some $u_\eta \in W^{\eta}$, where $W^{\eta}:=\big\{u \in W_0^{1,p}(\Omega) \ : \ \|u\|_{L^{\infty}(\Omega)} \leq \eta \big\}\,,$ and $\eta$ is the positive parameter given in (\ref{hnegativo})\,;
\item[$ii)$] $u_\eta \in [0, \delta]$,\, where $\delta$ is the positive parameter given in $($\ref{hnegativo}$)$$;$
\item[$iii)$] ${u}_\eta$ is a non-negative weak solution of problem$~($\ref{auxiliary}$)$.
\end{enumerate}
\end{lemma}

Fix $\tilde{\lambda}_0 \in (0, \ell_0)$ and define
$
K(x) := \tilde{\lambda}_0$ and $h(x, s, \lambda) := \lambda \beta(x)s^q+\tilde{\lambda}_0 s^{p-1} + f (s)$.
Using the fact that
$h(x,s,0)=\tilde{\lambda}_0s^{p-1}+f(s)\,,$
we deduce that there exist the sequences $\{\delta_j\}_j , \{\eta_j\}_j, \{s_j\}_j$ and $\{\lambda_j\}_j$ such that $\lambda_j>0$,
$0 < \eta_{j+1} < \delta_j < s_j < \eta_j<1$, ${\displaystyle \lim_{j\to +\infty} \eta_j = 0}$,
and $
h(x,s, \lambda) \leq 0$
a.e. $x\in \Omega$, for all $s\in [\delta_j,\eta_j]$, $\lambda\in [-\lambda_j, \lambda_j]$ and $j\in \NN$ large enough.

For any $j\in \NN$, we define
$
h_j(x,s, \lambda):=h(x,\tau_{\eta_j}(s), \lambda)$ and $H_j(x, s, \lambda) :=\int^s_0 h_j(x, t, \lambda) dt$,
for  $x\in \Omega$, $s\geq 0$ and $\lambda \in [-\lambda_j, \lambda_j]$.
By straightforward computation we deduce that $h_j$ satisfies all the assumptions of Lemma~\ref{thauxiliary} for $j$ large, with $\delta=\delta_j$ and $\eta=\eta_j$.
For any $j\in \NN$, let $\mathcal E_{j, \lambda}$ be the energy functional
$
\mathcal E_{j, \lambda}:=\mathcal E_{K, h_j(\cdot, \cdot,\lambda)}$.
By Lemma~\ref{thauxiliary}
we deduce that for $j$ sufficiently large and provided $|\lambda|\leq \lambda_j$, there exists $u_{j, \lambda} \in W^{\eta_j}$ such that
\begin{equation} \label{3.1.2ter}
\min_{u\in W^{\eta_j}}\mathcal E_{j, \lambda}(u)=\mathcal E_{j, \lambda}(u_{j, \lambda})
\end{equation}
\begin{equation} \label{3.1.3ter}
u_{j, \lambda} (x)\in [0,\delta_j] {\rm \,\,\,\,\, for \,\, a.e.} \,\, x\in \Omega,
\end{equation}
and
\begin{equation} \label{3.1.4ter}
u_{j, \lambda}\,\,\, \mbox{is a non-negative weak solution of (\ref{auxiliary}) with $h=h_j$}\,.
\end{equation}
Since for $j$ sufficiently large,
$
0\leq u_{j, \lambda}(x)\leq \delta_j<\eta_j
$
a.e. $x\in \Omega$, we have
$h_j(x, u_{j,\lambda}(x), \lambda)=h(x, u_{j,\lambda}(x), \lambda)\,,$
so that $u_{j, \lambda}$ is a non-negative weak solution of problem~(\ref{problem}), provided that $j$ is large and $|\lambda|\leq\lambda_j$.

It remains to prove that for any $k\in \NN$, problem~(\ref{problem}) admits at least $k$ distinct solutions, for suitable values of $\lambda$.
At this purpose, we note that for any $u\in W^{1,p}_0(\Omega)$
$$\begin{array}{ll}
\mathcal{E}_{j,\lambda}(u) & = \int_\Omega a(x,\nabla u(x))\,dx-\frac{\lambda}{q+1}\int_\Omega \beta(x)|u(x)|^{q+1}\,dx - \int_\Omega F(u(x))\,dx\\
& =\mathcal{E}_{j,0}(u) -\frac{\lambda}{q+1}\int_\Omega \beta(x)|u(x)|^{q+1}\,dx\,.
\end{array}$$

{\bf Claim.} There exists an increasing sequence $\{\theta_j\}_j$ such that $\theta_j< 0$, ${\displaystyle \lim_{j\to +\infty}\theta_j=0}$ and
$
\theta_{j-1}<\mathcal E_{j,0}(u_{j,0})<\theta_j
$
for $j\geq j^*$, with $j^*\in \NN$.

First, note that the function
$(x, s)\mapsto h(x,s,0)=\tilde \lambda_0s^{p-1}+f(s)$
verifies all the assumptions of Lemma~\ref{thauxorigin}. Thus, there exist $\ell > 0$ and $\zeta\in (0,\eta_1)$ such that
$
F(s) \geq -\, \ell s^p\,\,\, \mbox{for all} \,\,\, s\in (0,\zeta)
$
and
there is a sequence $\{\tilde s_j\}_j$ such that $0<\tilde s_j \to 0$ as $j\to +\infty$  such that
for all $L>0$,
$
F(s_j)> L s_j^p
$
for $j\in \NN$ large enough.
Also, since $\delta_j\searrow 0$ as $j\to +\infty$, we can choose a subsequence of $\{\delta_j\}_j$, still denoted by $\{\delta_j\}_j$, such that
$
\tilde s_j\leq \delta_j
$
for all $j\in \NN$.

Now, for any $s >0$ we need to define the function~$z_s$ as follows:
\begin{equation} \label{2.1.1}
z_s(x):=\left\{
\begin{array}{ll}
0 & \mbox{ if $x \in \Omega \setminus B(x_0,r)$} \\
\displaystyle\frac{2s}{r} \left(r- |x-x_0| \right)
& \mbox{ if $x \in B(x_0,r) \setminus B(x_0,{r}/{2})$} \\
s & \mbox{ if $x \in B(x_0,{r}/{2})$}\,,
\end{array}
\right.
\end{equation}
which is such that $z_s\geq 0$ in $\Omega$, $z_s \in W_0^{1,p}(\Omega)$ and $\|z_s\|_{L^{\infty}(\Omega)}=s$.
Here $x_0\in \Omega$ and $r>0$ is such that $B(x_0, r) \subset \Omega$
In the following we denote by $\tilde z_j:=z_{\tilde s_j}$.

Now, let us fix $j\in \NN$ sufficiently large. We have $
\mathcal E_{j,0}(u_{j,0})\leq \mathcal E_{j,0}(\tilde z_j)<0$ and
$$
\mathcal E_{j,0}(u_{j,0})  \geq -\int_\Omega F(u_{j,0}(x))\,dx
 \geq -\int_\Omega \int_0^{u_{j,0}(x)}|f(s)|\,ds\,dx
 \geq -\int_\Omega \int_0^{\delta_j}|f(s)|\,ds\,dx
 \geq d_j\,.
$$
Note that $\{c_j\}_j$ and $\{d_j\}_j$ are such that $d_j<c_j<0$ for any $j\in \NN$ and ${\displaystyle \lim_{j\to +\infty}c_j=\lim_{j\to +\infty}d_j=0}$.
 Thus, we can extract two subsequences, still denoted by $\{c_j\}_j$ and $\{d_j\}_j$, such that the above properties hold true and the sequences $\{c_j\}_j$ and $\{d_j\}_j$ are non-decreasing. Now, we define
$$\theta_j:=\left\{\begin{array}{ll}
c_j & \mbox{if $j\in \NN$ is even}\\
d_j & \mbox{if $j\in \NN$ is odd}\,.
\end{array}
\right.$$
We deduce that for $i$ large enough
$\theta_{2i-1}= d_{2i-1}\leq d_{2i}<\mathcal E_{2i,0}(u_{2i,0})<c_{2i}=\theta_{2i}$,
which proves the Claim.

\medskip
Now, for any $j\geq j^*$ let
\begin{equation}\label{}
\lambda'_j:=\frac{(q+1)(\mathcal E_{j,0}(u_{j,0})-\theta_{j-1})}{(\|\beta\|_{L^\infty(\Omega)}+1)\mathcal L(\Omega)}
\qquad \lambda''_j:=\frac{(q+1)(\theta_j-\mathcal E_{j,0}(u_{j,0}))}{\|\beta\|_{L^1(\Omega)}+1}\,.
\end{equation}
Note that $\lambda'_j$ and $\lambda''_j$ are strictly positive and they are independent of $\lambda$.
For any fixed $k\in \NN$, let
$$\Lambda_k:=\min\Big\{\lambda_{j^*+1}, \dots, \lambda_{j^*+k}, \lambda'_{j^*+1}, \dots, \lambda'_{j^*+k}, \lambda''_{j^*+1}, \dots, \lambda''_{j^*+k}\Big\}\,.$$
Of course, $\Lambda_k>0$ is independent of $\lambda$\,.
Also, if $|\lambda|\leq\Lambda_k$, then $|\lambda|\leq\lambda_j$ for any $j=j^*+1, \dots, j^*+k$\,.
As a consequence of this, for any $\lambda\in \RR$ with $|\lambda|\leq\Lambda_k$,
$u_{j,\lambda}$ is a non-negative weak solution of problem~(\ref{problem})
for any $j=j^*+1, \dots, j^*+k$.
Let us show that these solutions are distinct. At this purpose, note that $u_{j,\lambda}\in W^{\eta_j}$  and so
$
\mathcal E_{j, 0}(u_{j, 0})={\displaystyle\min_{u\in W^{\eta_j}}\mathcal E_{j, 0}(u)}\leq \mathcal E_{j, 0}(u_{j, \lambda})\,.$
Thus, for any $\lambda$ such that $|\lambda|\leq\Lambda_k$ we obtain
\begin{equation}\label{Ej1}
\begin{array}{ll}
\di\mathcal E_{j, \lambda}(u_{j, \lambda}) &\di = \mathcal E_{j, 0}(u_{j, \lambda})-\frac{\lambda}{q+1}\int_\Omega \beta(x)|u_{j, \lambda}(x)|^{q+1}\,dx
 \geq \mathcal E_{j, 0}(u_{j, 0})-\frac{|\lambda|}{q+1}\|\beta\|_{L^\infty(\Omega)}\eta_j^{q+1} \mathcal L(\Omega)\\
& \di\geq \mathcal E_{j, 0}(u_{j, 0})-\frac{\Lambda_k}{q+1}\|\beta\|_{L^\infty(\Omega)}\mathcal L(\Omega)
\geq \mathcal E_{j, 0}(u_{j, 0})-\frac{\lambda'_j}{q+1}\|\beta\|_{L^\infty(\Omega)}\mathcal L(\Omega)
 =\theta_{j-1}\,,
\end{array}
\end{equation}
for any $j=j^*+1, \dots, j^*+k$.
On the other hand, using the fact that $\|\tilde z_j\|_{L^\infty(\Omega)}=\tilde s_j\leq\delta_j< \eta_j<1$, for any $\lambda$ with $|\lambda|\leq\Lambda_k$ we deduce that
\begin{equation}\label{Ej2}
\begin{array}{ll}
\mathcal E_{j, \lambda}(u_{j, \lambda}) & =\min_{u\in W^{\eta_j}}\mathcal E_{j, \lambda}(u)
 \leq \mathcal E_{j, \lambda}(\tilde z_j)
 = \mathcal E_{j, 0}(\tilde z_j)-\frac{\lambda}{q+1}\int_\Omega \beta(x)|\tilde z_j(x)|^{q+1}\,dx\\
&\di \leq \mathcal E_{j, 0}(\tilde z_j)-\frac{\lambda}{q+1}\int_{\{x\in \Omega : \lambda \beta(x)<0\}} \beta(x)|\tilde z_j(x)|^{q+1}\,dx
 \leq \mathcal E_{j, 0}(\tilde z_j)-\frac{\lambda}{q+1}\int_{\{x\in \Omega : \lambda \beta(x)<0\}} \beta(x)\,dx\\
& \di\leq \mathcal E_{j, 0}(\tilde z_j)+\frac{|\lambda|}{q+1}\int_{\{x\in \Omega : \lambda \beta(x)<0\}} |\beta(x)|\,dx
 \leq \mathcal E_{j, 0}(\tilde z_j)+\frac{\lambda}{q+1}\int_\Omega |\beta(x)|\,dx\\
&\di \leq \mathcal E_{j, 0}(\tilde z_j)+\frac{\Lambda_k}{q+1}\|\beta\|_{L^1(\Omega)}
\leq \mathcal E_{j, 0}(\tilde z_j)+\frac{\lambda''_j}{q+1}\|\beta\|_{L^1(\Omega)}
 =\theta_j\,,
\end{array}
\end{equation}
for any $j=j^*+1, \dots, j^*+k$.

Hence, by (\ref{Ej1}), (\ref{Ej2}) and the properties of $\{\theta_j\}_j$ we deduce that for any $j=j^*+1, \dots, j^*+k$
\begin{equation}\label{Eifinale}
\theta_{j-1}<\mathcal E_{j, \lambda}(u_{j, \lambda})<\theta_j<0\,,
\end{equation}
which yields that
$\mathcal E_{1, \lambda}(u_{1, \lambda})< \dots < \mathcal E_{k, \lambda}(u_{k, \lambda})<0\,.$
Thus, the solutions $\Big\{u_{1, \lambda}, \dots, u_{k, \lambda}\Big\}$ are all distinct and non-trivial, provided that $|\lambda|\leq\Lambda_k$.

Finally, we estimate the $W^{1,p}_0$--norm of $u_{j,\lambda}$. For all $j=j^*+1, \dots, j^*+k$ and $|\lambda|\leq\Lambda_k$ we have
$$\begin{array}{ll}
\di\frac{\Gamma_1}{p}\|u_{j,\lambda}\|_{W^{1,p}_0(\Omega)}^p &\di \leq \mathcal E_{j,\lambda}(u_{j,\lambda})+\frac{\lambda}{q+1}\int_\Omega \beta(x)|u_{j,\lambda}(x)|^{q+1}\,dx + \int_\Omega F(u_{j,\lambda}(x))\,dx\\
& \di< \theta_j+\frac{|\lambda|}{q+1}\|\beta\|_{L^\infty(\Omega)}\delta_j^{q+1}+ \int_\Omega \int_0^{\delta_j}|f(s)|\,ds\,dx
 < \frac{\Lambda_k}{q+1}\|\beta\|_{L^\infty(\Omega)}\delta_j + \bar C\delta_j\,,
\end{array}$$
for a suitable positive constant $\bar C$. It follows that
$\|u_{j,\lambda}\|_{W^{1,p}_0(\Omega)}\leq \tilde C\delta_j^{1/p}\,,$
where $\tilde C>0$.
Since $\delta_j\to 0$ as $j \to +\infty$, without loss of generality, we may assume that
$\delta_j\leq \min\{\tilde C^{-p}, 1\} 1/j^p$,
and this gives
$\|u_{j,\lambda}\|_{W^{1,p}_0(\Omega)}\leq 1/j$
for all $j=j^*+1, \dots, j^*+k$, provided that $|\lambda|\leq\Lambda_k$.
This completes the proof of Theorem~\ref{thorigin}.\qed


\section{Oscillation at infinity}\label{subsec:infty}
In this section we assume that the nonlinear term $f$ satisfies the following assumptions.
\begin{equation}\label{f2infty}
\displaystyle\liminf_{s\to +\infty} \frac{f(s)}{s^{p-1}}=:-\ell_\infty\in[-\infty, 0)\,;
\end{equation}
\begin{equation}\label{f1infty}
-\infty < \displaystyle\liminf_{s\to +\infty} \frac{F(s)}{s^p}\leq \limsup_{s\to +\infty} \frac{F(s)}{s^p} = +\infty\,.
\end{equation}

A function satisfying these conditions is
$f(s)=\alpha s^{\alpha-1}\Big(1-\sin s^{\sigma}\Big)-\sigma s^{\alpha+\sigma-1} \cos s^{\sigma}-p\gamma s^{p-1}\,,$
where $\alpha$, $\sigma$ and $\gamma$ are such that $\alpha>p$, $\sigma>0$ and $\gamma>0$.


In this setting the counterpart of Theorem~\ref{thorigin} can be stated as follows.
\smallskip
\begin{theorem}\label{thinfty}
Assume that $f$ satisfies relations $($\ref{f2infty}$)$, $($\ref{f1infty}$)$, and $f(0)=0$. If either
\begin{enumerate}
\item[$a)$] $q = p-1$, $\ell_\infty\in (0, +\infty)$ and $\lambda \beta(x) < \lambda_{\infty}$ a.e. $x\in \Omega$ for some $\lambda_{\infty}\in(0, \ell_\infty)$ or
\item[$b)$] $q = p-1$, $\ell_\infty=+\infty$ and $\lambda \in \RR$ is arbitrary or
\item[$c)$] $0<q<p-1$ and $\lambda \in \RR$ is arbitrary$,$
\end{enumerate}
then there exists a sequence $\{u_j\}_j$ in  $W_0^{1,p}(\Omega)$ of distinct weak solutions of problem~$($\ref{problem}$)$ such that
$${\displaystyle \lim_{j\to +\infty} \|u_j\|_{L^{\infty}(\Omega)}=+\infty.}$$

 Assume that $q>p-1$. Then for every $k\in \NN$ there exists $\Lambda_k>0$ such that problem~$($\ref{problem}$)$ has at least $k$ distinct weak solutions $u_1, \dots, u_k\in W^{1,p}_0(\Omega)$
satisfying
$\|u_j\|_{L^{\infty}(\Omega)}\geq j-1$ for all $j=1,\dots, k$,
provided that $|\lambda|<\Lambda_k$.
\end{theorem}
\smallskip

We refer to \cite{morase} for the proof and several related results. We also refer to the marvelous recent book by Ciarlet \cite{ciarletsiam} for the rigorous qualitative analysis of many models described by nonlinear partial differential equations.

\medskip{\bf Acknowledgements.}
G. Molica Bisci and R. Servadei were supported by the GNAMPA Project 2013 {\it Problemi non-locali di tipo Laplaciano frazionario}, V.~R\u adulescu acknowledges the support through Grant CNCS PCE--47/2011, while R.~Servadei was supported by the MIUR National Research Project
{\it Variational and Topological Methods in the Study of Nonlinear
Phenomena} and by the ERC grant $\epsilon$ ({\it Elliptic PDE's and
Symmetry of Interfaces and Layers for Odd Nonlinearities}).


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