\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 162, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/162\hfil
A jumping problem for a quasilinear equation]
{A jumping problem for a quasilinear equation involving the
 1-Laplace operator}

\author[A. Canino, G. Riey \hfil EJDE-2011/162\hfilneg]
{Annamaria Canino, Giuseppe Riey}  % in alphabetical order

\address{Annamaria Canino \newline
Dipartimento di Matematica, UNICAL\\
Ponte Pietro Bucci 31B, 87036, Arcavacata di Rende,
Cosenza, Italy}
\email{canino@mat.unical.it}

\address{Giuseppe Riey \newline
Dipartimento di Matematica, UNICAL\\
Ponte Pietro Bucci 31B, 87036, Arcavacata di Rende,
Cosenza, Italy}
\email{riey@mat.unical.it}

\thanks{Submitted September 20, 2011. Published December 8, 2011.}
\subjclass[2000]{35J65, 49J40}
\keywords{Non-smooth analysis; jumping problems;
 hemivariational inequalities}

\begin{abstract}
 We prove the existence of solutions for a jumping problem
 of a functional whose principal part is the total variation.
 Our main tool is a nonsmooth variational method.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

The expression \emph{jumping problems} appeared first
in the celebrated paper by Ambrosetti and Prodi \cite{AmPr}.
They studied a semilinear elliptic PDE  of the form
\begin{equation}\label{eq Ambrosetti Prodi}
\begin{gathered}
- \Delta u + g(x,u) = w  \quad\text{in } \Omega\,,\\
u=0  \quad\text{on } \partial\Omega\,,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^n$,
$w\in W^{-1,2}(\Omega)$ and
$g:\Omega\times\mathbb{R}\to\mathbb{R}$ such that
$$
\lim_{s\to -\infty}\frac{g(x,s)}{s}=-\alpha,\quad
\lim_{s\to +\infty}\frac{g(x,s)}{s}=-\beta\,.
$$
Under suitable assumptions on $g$ and the condition
$\beta<\lambda_1<\alpha<\lambda_2$,
where $\lambda_1$ and $\lambda_2$ are the first two eigenvalues of
the operator $-\Delta$, the authors
provided a precise description of the number of solutions $u$,
of \eqref{eq Ambrosetti Prodi}, in dependence of $w$.

The result has been extended in various directions, also
by means of variational methods applied to the associated
functional $f:W^{1,2}_0(\Omega)\to\mathbb{R}$ defined as
$$
f(u)=\frac{1}{2}\,\int_\Omega |\nabla u|^2 +\int_\Omega G(x,u)
- \langle w,u\rangle\,,
$$
where $G(x,s)=\int_0^s g(x,t)\,dt$.
Let us mention, in particular, the case in which
$f:W^{1,2}_0(\Omega)\to\mathbb{R}$ is the defined as
$$
f(u)=\frac{1}{2}\,\int_\Omega\sum_{i,j=1}^n a_{ij}(x,u)D_i u D_j u
+\int_\Omega G(x,u) - \langle w,u\rangle\,,
$$
considered in~\cite{Ca1}, or, more generally, the case in which
$f:W^{1,p}_0(\Omega)\to\mathbb{R}$ is defined as
\begin{equation}
\label{eq:APp}
f(u)=\int_\Omega L(x,u,\nabla u) +\int_\Omega G(x,u)
- \langle w,u\rangle
\end{equation}
with
$$
\lim_{s\to -\infty}\frac{g(x,s)}{|s|^{p-2}s}=-\alpha,\quad
\lim_{s\to +\infty}\frac{g(x,s)}{|s|^{p-2}s}=-\beta\,,
$$
considered in \cite{GrSq1}.
In these extensions, the feature is that the functional $f$ is
continuous, but not locally Lipschitz, so that the
nonsmooth variational methods
of~\cite{CanDe,CoDeMa,DeMa} are used.
Further developments in this direction
are contained in~\cite{Ca2,GrSq2}.

Information about the number of solutions in dependence
of $w$ are given, in the case of~\eqref{eq:APp}, by setting
\[
w = t\varphi_1^{p-1} + w_0\,,
\]
where $\varphi_1$ is a positive first eigenfunction of the
$p$-Laplace operator and $w_0\in W^{-1,p'}(\Omega)$.
From a variational point of view, this is equivalent to study
the functional
\[
f_t(u)=\int_\Omega L(x,u,\nabla u) +\int_\Omega G(x,u)
- t\int_\Omega \varphi_1^{p-1}u - \langle w_0,u\rangle\,,
\]
in which the ``exploring term''
\[
u\mapsto \int_\Omega \varphi_1^{p-1}u
\]
has lower order at infinity with respect to the principal part
of the functional, as $p>1$.
Under the assumption that $\beta < \lambda_1 < \alpha$,
the main result asserts that there exist
$\underline t \leq \overline t$ such that the problem has at least
two solutions for $t\geq\overline t$ and no solution for
$t\leq\underline t$.

In this article we are interested in a corresponding result
for the case $p=1$.
At a naive level, we would consider the functional
$f_t:W^{1,1}_0(\Omega)\to\mathbb{R}$ defined as
\[
f_t(u)=\int_\Omega |\nabla u| +\int_\Omega G(x,u)
- t\int_\Omega \varphi H(u) \,,
\]
where $\varphi>0$ in $\Omega$ and
$H(s) = \int_0^s \frac{1}{\sqrt{1+t^2}}\,dt$.
The choice of
\[
u\mapsto \int_\Omega \varphi H(u)
\]
as ``exploring term'' is related to the need of considering
a lower order term at infinity with respect to the principal
part of the functional.

It is well known that direct variational methods do not work
properly in $W^{1,1}_0(\Omega)$, so that we will actually
consider a ``relaxed'' functional defined on $BV_0(\Omega)$, which is the subspace
of $BV(\mathbb{R}^n)$ made by functions vanishing outside $\Omega$.
Even after this first extension, (nonsmooth) critical point theory
cannot be directly applied, as the functional does not satisfy
the Palais-Smale condition.
This is due to the fact that such a condition requires a norm convergence,
which is almost impossible in $BV$, because of the lack strict
convexity of the principal part of the functional.
For this reason, as in~\cite{DeMag, DeMaRa}, we further extend
the relaxed functional to $L^{n/(n-1)}(\mathbb{R}^n)$ with value
$+\infty$ outside $BV_0(\Omega)$.
In this way the functional becomes only lower semicontinuous, but
the nonsmooth critical point theory of~\cite{CanDe,CoDeMa,DeMa}
can be successfully applied.
\par
The conditions involving $\alpha$ and $\beta$ will be expressed
again as $\beta<\lambda_1<\alpha$, where $\lambda_1$ is the first
eigenvalue of the $1$-Laplace operator as defined e.g.
in~\cite{KaSc}, and
$$
\liminf_{s\to-\infty}\,\frac{G(x,s)}{s} \geq \alpha\,,\quad
\liminf_{s\to+\infty}\,\frac{G(x,s)}{s} \geq - \beta\,.
$$
We will prove that there exists $\overline t$ such that
the problem has at least two solutions, provided that
$t\geq\overline t$.
A model case, which is covered in our result, is given by
\[
G(x,s) = -\beta s^+ - \alpha s^-\,.
\]
As $G$ is only locally Lipschitz, the concept of solution is
given in terms of ``hemivariational inequality'' as in
\cite{DeMaRa}.


The content of the paper runs as follows:
in Section \ref{sezione preliminari}
we recall some basic tools about functions with bounded variation and
non-smooth critical point theory and
in Sections \ref{sezione risultato principale}
and \ref{sezione dimostrazione risultato principale} we state and prove
our main result.

\section{Notation and preliminary results}\label{sezione preliminari}

Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$,
$n\geq 2$, with Lipschitz boundary.

\subsection{BV functions}

We denote by $BV(\Omega)$ the subspace of $L^1(\Omega)$ made
by functions whose distributional gradient $Du$ is a
vector-valued Radon measure with bounded variation.
For $u\in BV(\Omega)$, $|Du|$ denotes the total
variation of $Du$. We write $\nabla u$ for the
\emph{approximate differential}
of $u$ defined as in \cite[Definition 3.70]{AmFuPa}.
Denoted by $\mathcal L^n$ the Lebesgue measure in $\mathbb{R}^n$, we
can decompose $Du$ as: $Du=D^a u+D^s u$, where $D^a u$ and $D^s u$
are the absolute continuous part and the singular part of $Du$
with respect to $\mathcal L^n$.
It turns out that $D^a u=\nabla u\,d\mathcal L^n$.
For $u\in W^{1,1}(\Omega)$, the approximate differential of $u$
coincides with the distributional gradient of $u$
and we have: $Du=\nabla u\,d\mathcal L^n$.

\begin{remark}\label{bv norma} \rm
$BV(\Omega)$ is endowed with the norm:
$$
\|u\|_{BV(\Omega)}=|Du|(\Omega)+\int_\Omega |u|\,d\mathcal{L}^{n}\,.
$$
\end{remark}

\begin{definition}[Trace]\rm
Let $u\in BV(\Omega)$.
For $\mathcal{H}^{n-1}$-a.e. $x\in\partial\Omega$,
there exists $u^{\partial \Omega}(x)$ such that:
$$
\lim_{\rho\to 0^+}\frac{1}{\rho^n}
\int_{\Omega\cap B_\rho (x)}|u(y)-u^{\partial\Omega}(x)|\,
d\mathcal{L}^{n}(y)=0\,,
$$
where $B_\rho(x)$ is the open ball of radius $\rho$ and center $x$.
\end{definition}

\begin{definition}\rm
We set:
$BV_0(\Omega):=\{u\in BV(\mathbb{R}^n):
u=0\; \mathcal{L}^{n}\text{ a.e. in }\mathbb{R}^n\setminus\Omega\}$.
\end{definition}

Of course, $BV_0(\Omega)$ is a closed linear subspace of
$BV(\mathbb{R}^n)$.
Moreover, for every $u\in BV_0(\Omega)$ it turns out that
\[
|Du|(\overline\Omega) =
|Du|(\Omega)+\int_{\partial\Omega}|u^{\partial\Omega}|d\mathcal{H}^{n-1}\,.
\]

\begin{theorem}[Sobolev-type inequality in $BV$]\label{bv-poincare}
There exists $S>0$ such that, for every $u\in BV(\Omega)$, we have
$$
S\,\|u\|_{L^{\frac{n}{n-1}}(\Omega)}\leq
|Du|(\Omega)+\int_{\partial\Omega}|u^{\partial\Omega}|d\mathcal{H}^{n-1} \,.
$$
\end{theorem}

\begin{remark} \rm
Theorem \ref{bv-poincare} implies that, in $BV_0(\Omega)$,
$|Du|(\overline\Omega)$ is a norm equivalent to the canonical norm
of $BV(\mathbb{R}^n)$.
\end{remark}

\begin{theorem}\label{bv-immersioni}
Let $p$ belong to the interval $[1,\frac{n}{n-1}[$.
Then the inclusion of $BV_0(\Omega)$ in $L^p(\mathbb{R}^n)$ is compact.
\end{theorem}

The following theorem states the existence of the first eigenvalue
of the total variation.

\begin{theorem}\label{autovalore}
There exists
$$
\lambda_1=\min_{u\in BV_0(\Omega)\setminus\{0\}}
\frac{
{|Du|(\overline\Omega)}}{
{\int_\Omega |u|\,d\mathcal{L}^n}}\,.
$$
\end{theorem}

For more details on $BV$-functions and their properties see for
instance \cite{AmFuPa, KaSc}.
Let $\psi:\mathbb{R}^n\to\mathbb{R}$ be a convex function such that
there exists $M>0$ for which
$$
|p|\leq\psi(p)\leq M(1+|p|)\,.
$$
The functional
\begin{equation}\label{funz crescita lineare}
F(u)=\int_\Omega\psi(\nabla u)d\mathcal L^n
\end{equation}
is lower-semicontinuous in $W^{1,1}(\Omega)$
but, since this space is not reflexive, it has not good properties
for the compactness.
Therefore, when one deals with functionals with linear growth in the gradient,
it is usual to extend $F$ on the larger space $L^{n/(n-1)}(\Omega)$ in such a way
to have that this extension is lower-semicontinuous.
This procedure is called \emph{relaxation} and
on past years it was widely investigated
for functionals of the type in \eqref{funz crescita lineare}
and with more complicated integrands $\psi$, depending also
on $x$ and $u$ and eventually also under trace constraints
(see \cite{AmDa,AmFuPa,BoFoMa1,BoFoMa2}).
For $F$ defined in \eqref{funz crescita lineare}
its relaxation, denoted by $\overline F$,
takes the form
\begin{equation}\label{rilassato funz crescita lineare}
\overline F(u)=\begin{cases}
\int_\Omega\psi(\nabla u)d\mathcal L^n+
\int_\Omega \psi^\infty\big(\frac{dD^s u}{d|D^s u|}\big)d|D^s u|
&\text{if $u\in BV(\Omega)$}\,,\\
+\infty
&\text{if $u\in L^{n/(n-1)}(\Omega)\setminus BV(\Omega)$}\,,
\end{cases}
\end{equation}
where
$$
\psi^\infty(p)=\lim_{t\to +\infty}\frac{\psi(tp)}{t}\,.
$$

\begin{remark}\label{rilassato 1-lap}\rm
If $\psi(p)=|p|$, $F$ is the Diriclet functional
with linear growth
$$
F(u)=\int_\Omega|\nabla u|d\mathcal L^n
$$
and an easy computation gives
$$
\overline F(u)=\int_\Omega|\nabla u|d\mathcal L^n+
|D^s u|(\Omega)\equiv |Du|(\Omega)
\quad\text{if }u\in BV(\Omega)\,.
$$
If $F$ is defined only on $W^{1,1}_0(\Omega)$, then the
relaxed functional can be naturally identified with
$\overline F:L^{n/(n-1)}(\mathbb{R}^n)\to[0,+\infty]$ defined as
\[
\overline F(u)=\begin{cases}
|Du|(\overline\Omega)
&\text{if $u\in BV_0(\Omega)$}\,,\\
+\infty
&\text{if $u\in L^{n/(n-1)}(\mathbb{R}^n)\setminus BV_0(\Omega)$}\,.
\end{cases}
\]
\end{remark}

\begin{remark}\rm
In view of Remark \ref{rilassato 1-lap}
it is usual to refer to the number $\lambda_1$ given in
Theorem \ref{autovalore} as the first eigenvalue of
the $1$-Laplace operator with homogeneous Dirichlet condition.
\end{remark}

In \cite{An} it is computed the first variation for
functionals involving a term of the type in
\eqref{rilassato funz crescita lineare}
with a $\psi$ depending also on $x$.
For convex functionals of the same type and involving also $u$,
a characterization of the subdifferential is performed in \cite{AnCaMa}.
In Section \ref{sezione risultato principale}
we consider non-convex functionals containing also
a non locally Lipschitz term.
In this case the critical points can be characterized by means
of hemivariational inequalities, using subdifferential calculus and
nonsmooth analysis (see \cite{CamDe, DeMaRa}).


\subsection{Nonsmooth critical point theory}

Let $X$ be a real Banach space and let $X^*$ be its dual space.
First of all, let us recall from~\cite{Cl} some basic notions.

\begin{definition}\rm
If $f:X\to\mathbb{R}$ is a locally Lipschitz function, we set,
for every $u,v\in X$,
\begin{equation}\label{derivata generalizzata}
f^{\circ}(u;v):=\limsup_{ z\to u ,\,w\to v,\,t\to 0^+}
\frac{f(z+tw)-f(z)}{t}\,.
\end{equation}
The real number $f^{\circ}(u;v)$ is called the
\emph{generalized directional derivative} of $f$ at $u$
with respect to the direction $v$.
For every $u\in X$, let also
\begin{equation}
\partial f(u)=\{u^*\in X^*:\,\,
\langle u^*,v\rangle\leq f^{\circ}(u;v)\,,\,\,
\forall v\in X\}\,.
\end{equation}
The set $\partial f(u)$ is called the
\emph{subdifferential} of $f$ at $u$.
\end{definition}

It turns out that $f^{\circ}$ is positively one-homogeneous
with respect to the second variable.

Assume now that
$f:X\to]-\infty,+\infty]$ is a lower semicontinuous function and set
\[
\operatorname{epi}(f)=\{(u,\lambda)\in X\times\mathbb{R}:\,f(u)\leq\lambda\}\,.
\]
\begin{definition}\label{defn:defnws}\rm
For every $u\in X$ with $f(u)<+\infty$, we denote by
$|d f|(u)$ the supremum of the
$\sigma$'s in $[0,+\infty[$ such that there exist
a neighborhood $W$ of $(u,f(u))$ in $\operatorname{epi}(f)$,
$\delta>0$ and a continuous map
$\mathcal{H}:W\times[0,\delta]\to X$
satisfying
\[
\|\mathcal{H}((w,\mu),t)-w\| \leq  t\,,\quad
f(\mathcal{H}((w,\mu),t)) \leq  \mu - \sigma t\,,
\]
whenever $(w,\mu)\in W$ and $t\in[0,\delta]$.

The extended real number $|d f|(u)$ is called {\em the weak slope}
of $f$ at $u$.
\end{definition}

The above notion has been introduced in \cite{DeMa}, following
an equivalent approach.
The version we have recalled here is taken from \cite{CamDe}.
According to \cite{DeMaTo}, we also define a function
$\mathcal{G}_{f}:\operatorname{epi}(f)\to\mathbb{R}$ by
$\mathcal{G}_{f}(u,\lambda)=\lambda$.

\begin{definition}\rm
A point $u\in X$ is said to be a
\emph{lower critical point} of $f$,
if $f(u)<+\infty$ and $|d f|(u)=0$.
\end{definition}

\begin{definition}[Cerami-Palais-Smale condition] \rm
Let $c\in\mathbb{R}$.
A sequence $(u_k)$ in $X$ is said to be a
\emph{Cerami-Palais-Smale sequence at level $c$} for $f$
($(CPS)_c$-sequence, for short),
if $f(u_k)\to c$ and $(1+\|u_k\|)|d f|(u_k) \to 0$.
We say that $f$ satisfies the
\emph{Cerami-Palais-Smale condition at level $c$}
($(CPS)_c$-condition, for short),
if every $(CPS)_c$-sequence for $f$ admits a strongly convergent
subsequence in $X$.
\end{definition}

\begin{definition}\rm
Let $c\in\mathbb{R}$. We say that $f$ satisfies condition $(epi)_c$,
if there exists $\varepsilon >0$ such that
\[
\inf\{|d \mathcal{G}_{f}|(u,\lambda):\,f(u)<\lambda,\,
|\lambda-c| < \varepsilon \} > 0 \,.
\]
\end{definition}

The next result is an extension of the celebrated
Mountain pass theorem~\cite{AmRa} to our setting.
For the proof, when the usual Palais-Smale condition is assumed,
we refer the reader to~\cite{CoDeMa, DeSc}.
We also refer to~\cite{Co} for the fact that the
Cerami-Palais-Smale condition can be reduced to the
Palais-Smale condition by a change of metric.

\begin{theorem}[Mountain pass theorem]\label{passo montano}
Let $u_0,u_1,\overline{u}\in X$ and $r>0$ be such that:
\begin{equation}
\|u_0-\overline{u}\|<r,\quad
\|u_1-\overline{u}\|>r
\end{equation}
and
\begin{equation}
\inf\{
f(u): u\in X,\, \|u-\overline{u}\|=r\}
\geq \max\{f(u_0),f(u_1)\}\,.
\end{equation}
Set
\[
\Gamma:=\{\gamma\in C([0,1];X): \gamma(0)=u_0,\, \gamma(1)=u_1\}
\]
and define
\[
c_1:=\inf_{u\in\overline{B_r(\overline{u})}}f(u),\quad
c_2=\inf_{\gamma\in\Gamma}\sup_{t\in [0,1]}f(\gamma(t))\,,
\]
so that
\[
c_1 \leq \inf\{f(u): u\in X,~\|u-\overline{u}\|=r\}
\leq c_2\,.
\]
Assume that $c_1>-\infty$, $c_2<+\infty$ and that
$f$ satisfies $(CPS)_c$ and $(epi)_c$ for $c=c_1, c_2$.

Then $f$ admits two distinct lower critical points $w_1, w_2$
with $f(w_1)=c_1$ and $f(w_2)=c_2$.
\end{theorem}

\section{Statement of the main result}
\label{sezione risultato principale}

We consider:
\begin{itemize}
\item $\varphi\in L^n(\Omega)$ such that $\varphi > 0$ a.e.
 in $\Omega$;
\item $G:\Omega\times\mathbb{R}\to \mathbb{R}$ such that
%
\begin{equation}\label{g1}
G(\cdot,s)\text{ is measurable for any $s\in\mathbb{R}$;}
\end{equation}
for every $t>0$ there exists $a_t\in L^1(\Omega)$ such that
\begin{equation} \label{g2}
|G(x,s_1)-G(x,s_2)|\leq a_t(x)|s_1-s_2|
\end{equation}
\\
for a.e. $x\in\Omega$ and every $s_1,s_2\in\mathbb{R}$ with
$|s_1|\leq t,\, |s_2|\leq t$;
there exist $\alpha,\beta\in\mathbb{R}$ such that
$\beta<\lambda_1<\alpha$ and
\begin{equation}\label{g5}
\liminf_{s\to-\infty} \frac{G(x,s)}{s} \geq \alpha\,,\quad
\liminf_{s\to+\infty} \frac{G(x,s)}{s} \geq - \beta
\quad\text{for a.e. $x\in\Omega$}\,.
\end{equation}
\end{itemize}

From \eqref{g2} it follows that $G(x,\cdot)$ is locally Lipschitz
for a.e. $x\in\Omega$. We denote by $G^{\circ}(x,s;t)$ the generalized
directional derivative with respect to the second variable.
Then we also assume that
\begin{itemize}
\item there exist $b_0\in L^1(\Omega)$ and $b_1\in L^n(\Omega)$
such that:
\begin{gather}
\label{g3} |G(x,s)|\leq b_1(x)|s|\quad
\text{for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$}\,,\\
\label{g4}
G^{\circ}(x,s;-s)\leq b_0(x) + b_1(x)|s| \quad
\text{for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$}\,,\\
\label{g6}
G^{\circ}(x,s;s)\leq b_0(x) + G(x,s) \quad
\text{for a.e. $x\in\Omega$ and every $s\in\mathbb{R}$ with
$s\leq 0$}\,.
\end{gather}
\end{itemize}

From \eqref{g3} it follows that the functional
\[
\{u \mapsto \int_\Omega G(x,u)\}
\]
is continuous on $L^{n/(n-1)}(\Omega)$,
although it is not locally Lipschitz, as there is no upper
bound for $G^{\circ}(x,s;s)$ when $s>0$.
In particular, the assumptions on $G$ are satisfied in the model
case
\[
G(x,s) = - \beta s^+ - \alpha s^-\,.
\]
We are interested in solutions $u\in BV_0(\Omega)$ of the
hemivariational inequality (see~\cite{DeMaRa})
\begin{equation}\label{eq:sol}
|Dv|(\overline\Omega) - |Du|(\overline\Omega)
 + \int_\Omega G^{\circ}(x,u;v-u)
\geq t\int_\Omega \varphi\frac{1}{\sqrt{1+u^2}}(v-u)\,,
\quad\forall v\in BV_0(\Omega)
\end{equation}
associated with the lower semicontinuous functional
$f_t:L^{n/(n-1)}(\mathbb{R}^n)\to\mathbb{R}\cup\{+\infty\}$ defined as:
\begin{equation}
f_t(u)=\begin{cases}
|Du|(\overline\Omega)+\int_\Omega G(x,u)
 -t\int_\Omega \varphi H(u) & \text{if } u\in BV_0(\Omega)\,,\\
+\infty & \text{if } u\in L^{n/(n-1)}(\mathbb{R}^n)\setminus
 BV_0(\Omega)\,,
\end{cases}
\end{equation}
where
\[
H(s)=\int_0^s\frac{1}{\sqrt{1+t^2}}\,dt\,.
\]
If $G(x,\cdot)$ is of class $C^1$ for a.e. $x\in\Omega$ and
we set $g(x,s)=D_sG(x,s)$, then
assumptions \eqref{g4} and \eqref{g6} are equivalent to
\begin{gather*}
s g(x,s)\geq - b_0(x) - b_1(x)|s|\,, \\
s g(x,s)\leq b_0(x) + G(x,s)\quad\text{for $s\leq 0$}\,.
\end{gather*}

\begin{remark}\rm
In view of Remark \ref{rilassato 1-lap} we recall
that $f_t$ is achieved by relaxation of the functional
defined on $W^{1,1,}_0(\Omega)$ as
$$
\int_\Omega |\nabla u|+\int_\Omega G(x,u)-t\int_\Omega\varphi H(u)\,.
$$
\end{remark}

\begin{theorem}\label{teorema principale}
There exists $\overline t\in\mathbb{R}$ such that,
for every $t\geq \overline t$, there exists at least
two solutions of~\eqref{eq:sol}.
\end{theorem}

\section{Proof of the main result}
\label{sezione dimostrazione risultato principale}

\subsection{Compactness properties}

\begin{proposition}\label{CPSc}
For every $t > 0$ and $c\in\mathbb{R}$, the functional $f_t$ satisfies
$(CPS)_c$.
\end{proposition}

\begin{proof}
First of all, by \eqref{g3} we have that $G(x,0)=0$.
Let $(u_k)$ be a sequence in $BV_0(\Omega)$ satisfying
\begin{gather}\label{CePS1}
\lim_{k\to+\infty}f_t(u_k)=c\,, \\
\lim_{k\to+\infty}(1+\|u_k\|_{n/(n-1)})|d f_t|(u_k)=0\,.
\end{gather}
By \cite[Theorems 2.16, 3.6 and 4.1]{DeMaRa}, there exists a sequence
$(w_k)$ in $L^n(\mathbb{R}^n)$ such that
\begin{equation}\label{CePS2}
\lim_{k\to+\infty}(1+\|u_k\|_{n/(n-1)})\|w_k\|_n=0\,,
\end{equation}
\begin{equation} \label{CePS3}
\begin{split}
 |Dv|(\overline\Omega)
&\geq  |Du_k|(\overline\Omega)-\int_\Omega G^{\circ}(x,u_k;v-u_k)\\
&\quad +t\int_\Omega\varphi\frac{1}{\sqrt{1+u_k^2}}(v-u_k)
 +\int_\Omega w_k(v-u_k)
 \quad\forall v\in BV_0(\Omega).
\end{split}
\end{equation}
In particular, the choice $v=u_k - u_k^-$ yields
\begin{align*}
&|Du_k|(\overline\Omega) + |Du_k^-|(\overline\Omega) \\
&\geq  |D(u_k-u_k^-)|(\overline\Omega) \\
&\geq  |Du_k|(\overline\Omega)-\int_\Omega G^{\circ}(x,-u_k^-;-u_k^-)
-t\int_\Omega\varphi\frac{1}{\sqrt{1+u_k^2}}\,u_k^-
 -\int_\Omega w_k u_k^-\,,
\end{align*}
hence
\[
|Du_k^-|(\overline\Omega) \geq
-\int_\Omega G^{\circ}(x,-u_k^-;-u_k^-)
-t\int_\Omega\varphi\frac{1}{\sqrt{1+u_k^2}}\,u_k^-
 -\int_\Omega w_k u_k^-\,,
 \]
which  by~\eqref{g6} implies
\begin{equation}
\begin{split}
f_t(u_k)
&\geq  |Du_k^+|(\overline\Omega)
+ \int_\Omega G(x,u_k^+)
+ \int_\Omega[G(x,-u_k^-) - G^{\circ}(x,-u_k^-;-u_k^-)]\\
&\quad -t\int_\Omega\varphi H(u_k)
-t\int_\Omega\varphi\frac{1}{\sqrt{1+u_k^2}}\,u_k^-
 -\int_\Omega w_k u_k^-\\
&\geq  |Du_k^+|(\overline\Omega)
+ \int_\Omega G(x,u_k^+) - \int_\Omega b_0
-t\int_\Omega\varphi H(u_k) - t\int_\Omega\varphi \\
&\quad - \int_\Omega w_k u_k^-.
\end{split}\label{eq:ft}
\end{equation}
By~\cite[Theorems 3.12 and 4.1]{DeMaRa}, it sufficies to prove
that $(u_k)$ is bounded in
$L^{n/(n-1)}(\mathbb{R}^n)$.

We first show that $(|Du_k^+|(\overline\Omega))$ is bounded.
Assume, for a contradiction, that
$\lim_{k\to+\infty}|Du_k^+|(\overline\Omega)=+\infty$
and consider the sequence $v_k:=u_k^+/\varrho_k$,
where $\varrho_k=|Du_k^+|(\overline\Omega)$.
Then, up to a subsequence, $(v_k)$ is convergent to some
$v\in BV_0(\Omega)$ with $|Dv|(\overline\Omega)\leq 1$
weakly in $L^{n/(n-1)}(\mathbb{R}^n)$ and a.e. in $\Omega$.
Moreover, we also have that $u_k^+(x)\to +\infty$ a.e. on
$\{v(x)\neq 0\}$.
Since $H(u_k) \leq H(u_k^+)$, from~\eqref{eq:ft} we infer that
\begin{equation}\label{CePS4}
\begin{split}
\frac{f_t(u_k)}{\varrho_k}
&\geq 1 + \int_\Omega \frac{G(x,\varrho_kv _k)}{\varrho_k} -
\frac{1}{\varrho_k}\int_\Omega b_0
-t\int_\Omega\varphi \frac{H(\varrho_kv _k)}{\varrho_k}\\
&\quad - \frac{t}{\varrho_k}\int_\Omega\varphi
- \frac{1}{\varrho_k}\int_\Omega w_k u_k^- \,.
\end{split}
\end{equation}
On the other hand, by~\eqref{g3} and~\eqref{g5} we have
\begin{gather*}
\frac{G(x,\varrho_k v _k)}{\varrho_k}
-t \varphi \frac{H(\varrho_kv _k)}{\varrho_k}
\geq - b_1(x)v_k - t \varphi v_k\,,
\\
\liminf_{k\to+\infty} \Big(\frac{G(x,\varrho_kv _k)}{\varrho_k}
-t \varphi \frac{H(\varrho_kv _k)}{\varrho_k}\Big)
\geq - \beta v\,.
\end{gather*}
From Fatou's lemma, \eqref{CePS1}, \eqref{CePS2}
and~\eqref{CePS4} we infer that
\[
0 \geq 1 - \beta \int_\Omega v
\geq 1 - \frac{\beta}{\lambda_1}|Dv|(\overline\Omega)
\geq 1 - \frac{\beta}{\lambda_1}\,.
\]
Since $\beta<\lambda_1$, a contradiction follows.
Therefore, $(|Du_k^+|(\overline\Omega))$ is bounded.

We now show that also $(|Du_k^-|(\overline\Omega))$ is bounded.
Again, we assume by contradiction that
$\lim_{k\to+\infty}|Du_k^-|(\overline\Omega)=+\infty$ and
consider the sequence $\tilde v_k:=u_k^-/\tilde\varrho_k$,
where $\tilde\varrho_k=|Du_k^-|(\overline\Omega)$.
Then, up to a subsequence, $(\tilde v_k)$ is convergent to some
$\tilde v\in BV_0(\Omega)$ with $|D\tilde v|(\overline\Omega)\leq 1$
weakly in $L^{n/(n-1)}(\mathbb{R}^n)$ and a.e. in $\Omega$.
Moreover, we have that $u_k^-(x)\to +\infty$ a.e. in
$\{\tilde v(x)\neq 0\}$.
On the other hand, from~\eqref{eq:ft} we also infer that
\begin{align*}
f_t(u_k)
&\geq |Du_k^+|(\overline\Omega)
+ \int_\Omega G(x,u_k^+) - \int_\Omega b_0
- t\int_\Omega\varphi H(u_k^+) \\
&\quad + t\int_\Omega\varphi H(u_k^-)
- t\int_\Omega\varphi  -\int_\Omega w_k u_k^-\,.
\end{align*}
Since $(|Du_k^+|(\overline\Omega))$ is bounded, from~\eqref{CePS1},
\eqref{CePS2} and \eqref{g3} we deduce that $(\varphi H(u_k^-))$
is bounded in $L^1(\Omega)$.
By Fatou's lemma it follows that $\tilde v=0$ a.e. in $\Omega$.
By~\eqref{g3} we have
\[
\frac{f_t(u_k)}{\tilde\varrho_k} \geq
\frac{1}{\tilde\varrho_k}\,|Du_k^+|(\overline\Omega) + 1 +
\frac{1}{\tilde\varrho_k}\,\int_\Omega G(x,u_k^+)
- \int_\Omega b_1v_k - t\,\int_\Omega\varphi v_k\,.
\]
Passing to the limit as $k\to\infty$ and taking into
account~\eqref{CePS1}, we infer that $0\geq 1$
and a contradiction follows.
Therefore also $(|Du_k^-|(\overline\Omega))$ is bounded.
In particular, $(u_k)$ is bounded in $L^{n/(n-1)}(\mathbb{R}^n)$
and the assertion follows.
\end{proof}

\subsection{Geometric conditions}

To apply Theorem \ref{passo montano}
we check that the functional $f_t$ satisfies the property stated
in the two following propositions.
\begin{proposition}\label{prop:locmin}
Set
\begin{gather*}
X^+:=\{u\in L^{n/(n-1)}(\mathbb{R}^n):
\text{$u\geq 0$ a.e. in $\mathbb{R}^n$},\\
B_r(v):=\{u\in L^{n/(n-1)}(\mathbb{R}^n):
\|u-v\|_{n/(n-1)} < r\}\,.
\end{gather*}
Then there exists $\overline t>0$ such that,
for every $t\geq \overline t$, there exist
$u_0\in X^+\cap BV_0(\Omega)$ and $r>0$
such that $f_t(u)\geq f_t(u_0)$ for every
$u\in \overline{B_r(u_0)}$.
\end{proposition}

\begin{proof}
From assumptions~\eqref{g5}, \eqref{g3} and
the fact that $\varphi\in L^n(\Omega)$, we infer
that $f_t$ is coercive on $X^+$, although it
does not on the whole $L^{n/(n-1)}(\mathbb{R}^n)$.
Moreover by assumptions on $G$ it is also lower semi-continuous
and therefore it immediately follows that there exists a minimum
point $u_0$ of $f_t$ on $X^+$ with $u_0\in BV_0(\Omega)$
and $f_t(u_0)\leq 0$.

Observe that for $t>0$,
\[
\{u\in BV_0(\Omega):|Du|(\overline\Omega) \leq 1\,,\,
u\geq 0\text{ a.e. in }
\Omega\,,\,
1 - \int_\Omega b_1 u + t\int_\Omega\varphi u \leq 0\}
\]
is a decreasing family of weakly compact subsets of
$L^{n/(n-1)}(\mathbb{R}^n)$ with empty intersection,
as $\varphi>0$ a.e. in $\Omega$.
Therefore there exists $\overline t>0$ such that
\begin{equation}\label{eq:to}
\{u\in BV_0(\Omega): |Du|(\overline\Omega) \leq 1\,,\,
u\geq 0\text{ a.e. in }
\Omega\,,\,
1 - \int_\Omega b_1 u +
\overline t\int_\Omega\varphi u \leq 0\} = \emptyset\,.
\end{equation}
Let us show that, for every $t\geq \overline t$,
$u_0$ is a local minimum of $f_t$ in $L^{n/(n-1)}(\mathbb{R}^n)$.
By contradiction, let $(v_k)$ be a sequence convergent to $u_0$
with $f_t(v_k) < f_t(u_0)$.
Since $G(x,0)=0$ and $H$ is an odd function,
by~\cite[Theorem~3.99]{AmFuPa} and~\eqref{g3} we have:
\begin{align*}
 f_t(v_k)
&=|D(v_k^+)|(\overline\Omega)+|D(v_k^-)|(\overline\Omega)
 + \int_\Omega G(x,v_k^+) + \int_\Omega G(x,-v_k^-) \\
&\quad- t\int_\Omega\varphi H(v_k^+) + t\int_\Omega\varphi H(v_k^-)\\
&\geq  f_t(u_0)+ |D(v_k^-)|(\overline\Omega)
 - \int_\Omega b_1 v_k^- + t\int_\Omega\varphi H(v_k^-)\\
&> f_t(v_k)+ |D(v_k^-)|(\overline\Omega)
 - \int_\Omega b_1 v_k^- + t\int_\Omega\varphi H(v_k^-)\,.
\end{align*}
It follows
\[
|D(v_k^-)|(\overline\Omega)
- \int_\Omega b_1 v_k^- + t\int_\Omega\varphi H(v_k^-) < 0\,.
\]
Since $v_k^-\to 0$ a.e. in $\Omega$, if we define
a sequence $(\eta_k)$ in $L^\infty(\Omega)$ by
\[
\eta_k = \begin{cases}
H(v_k^-) / v_k^- &\text{where $v_k^-\neq 0$}\,,\\
1 & \text{where $v_k^- = 0$}\,,
\end{cases}
\]
by Lebesgue theorem we infer that
\[
\lim_k \varphi\eta_k = \varphi
\quad\text{strongly in $L^n(\Omega)$}\,.
\]
Let $w_k = v_k^-/|D(v_k^-)|(\overline\Omega)$.
Then, up to a subsequence, $(w_k)$ is weakly convergent
in $L^{n/(n-1)}(\mathbb{R}^n)$ to some $w\in BV_0(\Omega)$
satisfying $w\geq 0$, $|Dw|(\overline\Omega)\leq 1$ and
\[
1 - \int_\Omega b_1 w + t\int_\Omega\varphi w \leq 0\,.
\]
This fact contradicts~\eqref{eq:to} and the assertion follows.
\end{proof}

\begin{proposition}\label{prop:-infty}
Let $\psi\in BV_0(\Omega)\setminus\{0\}$ be a first eigenfunction of the total
variation, so that $\lambda_1\int_\Omega|\psi|=|D\psi|(\overline\Omega)$,
with $\psi\geq 0$ a.e. in $\Omega$ (see~\cite{KaSc}).
Then, for every $t\in\mathbb{R}$, there holds:
\begin{equation}
\lim_{s\to +\infty}f_t(-s\psi)=-\infty\,.
\end{equation}
\end{proposition}

\begin{proof}
By the definition of $H$, \eqref{g5} and \eqref{g3} we have
\begin{gather*}
\limsup_{s\to+\infty}\frac{G(x,-s\psi(x))-t\varphi(x)H(-s\psi(x))}{s}
= - \psi(x)\liminf_{\sigma\to-\infty} \,\frac{G(x,\sigma)}{\sigma}
\leq - \alpha\psi(x)\,,
\\
\frac{G(x,-s\psi(x))-t\varphi(x)H(-s\psi(x))}{s}
\leq b_1(x)\psi(x) + |t|\varphi(x)\psi(x)\,.
\end{gather*}
From Fatou's lemma we infer that
\[
\limsup_{s\to+\infty}\frac{\int_\Omega(
G(x,-s\psi(x))-t\varphi(x)H(-s\psi(x)))}{s}
\leq - \alpha\int_\Omega \psi(x)
= - \frac{\alpha}{\lambda_1}\,|D\psi|(\overline\Omega) \,.
\]
Since $\alpha>\lambda_1$, the assertion follows.
\end{proof}

\subsection*{Conclusion}

Let $\overline t>0$ be as in Proposition~\ref{prop:locmin},
let $t\geq \overline t$ and let $u_0$ be a local minimum of $f_t$.
If we set $\overline u=u_0$, by Proposition~\ref{prop:-infty}
we can find $r>0$ and $u_1$ as in Theorem~\ref{passo montano}.
By Proposition~\ref{CPSc}, the functional $f_t$ satisfies $(CPS)_c$
for every $c\in\mathbb{R}$.
Moreover, by~\cite[Theorems~3.11 and 4.1]{DeMaRa} also $(epi)_c$ holds
for any $c\in\mathbb{R}$.
From Theorem~\ref{passo montano} we get the existence of at least two
lower critical points of $f_t$.
By~\cite[Theorems 2.16, 3.6 and 4.1]{DeMaRa}, they are solutions
of~\eqref{eq:sol}.

\begin{thebibliography}{00}

\bibitem{AmPr} A. Ambrosetti, G. Prodi;
\emph{On the inversion of differentiable maps with singularities
between Banach spaces},
Ann. Mat. Pura Appl. {\bf 93} (1972), 231--247.

\bibitem{AmRa} A. Ambrosetti, P.~H. Rabinowitz;
\emph{Dual variational methods in critical point theory and
applications}, J. Functional Analysis {\bf 14} (1973), 349--381.

\bibitem{AmDa} L. Ambrosio, G. Dal Maso;
\emph{On the relaxation in $BV(\Omega;\mathbb{R}^m)$ of quasi-convex
integrals},
J. Funct. Anal. {\bf 58} (1984), 225--253.

\bibitem{AmFuPa} L. Ambrosio, N. Fusco, D. Pallara;
\emph{Functions of Bounded Variation and Free Discontinuity Problems},
Clarendon Press, Oxford (2000).

\bibitem{AnCaMa} F. Andreu, V. Caselles, J. M. Mazon;
\emph{A strongly degenerate quasilinear equation: the elliptic case},
Ann. Scuola Norm. Pisa Cl. Sci. (5) Vol. III (2004), 555--587.

\bibitem{An} G. Anzellotti;
\emph{The Euler equation for functionals with linear growth},
Trans. American Math. Soc. {\bf 290} (1985), no. 2, 483--501.

\bibitem{BoFoMa1} G. Bouchitt\'e, I. Fonseca, L. Mascarenhas;
\emph{A global method for relaxation},
Arch. Rational Mech. Anal. {\bf 145} (1998), 51--98.

\bibitem{BoFoMa2} G. Bouchitt\'e, I. Fonseca, L. Mascarenhas;
\emph{Relaxation of variational problems under trace constraints},
Nonlinear Analysis {\bf 49} (2002), 221--246.

\bibitem{CamDe}
I. Campa, M. Degiovanni;
\emph{Subdifferential calculus and nonsmooth critical point theory},
SIAM J. Optim. {\bf 10} (2000), 1020--1048.

\bibitem{Ca1} A. Canino,
\emph{On a jumping problem for quasilinear elliptic equations},
Math. Z. {\bf 226} (1997), 573--589.

\bibitem{Ca2} A. Canino;
\emph{On the existence of three solutions for jumping problems
involving quasilinear operators},
Topol. Methods Nonlinear Anal. {\bf 18} (2001), no. 1, 1--16

\bibitem{CanDe} A. Canino, M. Degiovanni;
\emph{Nonsmooth critical point theory and quasilinear elliptic
equations, Topological Methods}
in DiŽerential Equations and Inclusions, A. Granas, M. Frigon,
G. Sabidussi (eds.), Montreal (1994), Kluwer Publishers, NATO
 ASI Series, Math. Phys.
Sci. 472 (1995), 1--50

\bibitem{Cl} F.~H. Clarke;
\emph{Optimization and nonsmooth analysis}, Canadian
Mathematical Society Series of Monographs and Advanced Texts,
John Wiley \& Sons Inc., New York, 1983.

\bibitem{Co} J.-N. Corvellec;
\emph{Quantitative deformation theorems and critical point theory},
Pacific J. Math. {\bf 187} (1999), 263--279.

\bibitem{CoDeMa} J.-N. Corvellec, M. Degiovanni, M. Marzocchi;
\emph{Deformation properties for continuous functionals and critical
point theory}, Topol. Methods Nonlinear Anal. {\bf 1} (1993), 151--171.

\bibitem{DeMaTo} E. De~Giorgi, A. Marino, M. Tosques;
\emph{Problemi di evoluzione in spazi metrici e curve di massima
pendenza},
Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)
{\bf 68} (1980), 180--187.

\bibitem{DeMag} M. Degiovanni, P. Magrone;
\emph{Linking solutions for quasilinear equations at critical
growth involving the ``$1$-Laplace'' operator},
Calc. Var. Partial Differential Equations
{\bf 36} (2009), 591--609.

\bibitem{DeMa}M. Degiovanni, M. Marzocchi;
\emph{A critical point theory for nonsmooth functionals},
Ann. Mat. Pura Appl. (4) {\bf 167} (1994), 73--100.

\bibitem{DeMaRa} M. Degiovanni, M. Marzocchi, V. D. Radulescu;
\emph{Multiple solutions of hemivariational inequalities with
area-type term},
Calc. Var. Partial Differential Equations {\bf 10} (2000), 355--387.

\bibitem{DeSc} M. Degiovanni, F. Schuricht;
\emph{Buckling of nonlinearly elastic rods in the presence of
obstacles treated by nonsmooth critical point theory},
Math. Ann. {\bf 311} (1998), 675--728.

\bibitem{GrSq1} A. Groli, M. Squassina;
\emph{On the existence of two solutions for a general class of
jumping problems},
Topol. Methods Nonlinear Anal. {\bf 21} (2003), 325--344.

\bibitem{GrSq2} A. Groli, M. Squassina;
\emph{Jumping Problems for Fully Nonlinear Elliptic
Variational Inequalities},
J. Convex Analysis {\bf 8} (2001), No. 2, 471--488

\bibitem{KaSc} B. Kawohl, F. Schuricht;
\emph{Diriclet problems for the 1-Laplace operator,
including the eigenvalue problem},
Commun. Contemp. Math. {\bf 9} (2007), 515--543.

\end{thebibliography}

\end{document}