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\title[Variational analysis on the Sierpi\'nski gasket]
{Variational analysis for a nonlinear elliptic problem on the Sierpi\'nski gasket}
%{Nonlinear Dirichlet problems on the Sierpi\'nski gasket}

\author{Gabriele Bonanno, Giovanni Molica Bisci and Vicen\c{t}iu R\u{a}dulescu}
\address[G. Bonanno]{Department of Science for Engineering and Architecture (Mathematics Section)
Engineering Faculty, University of Messina,
98166 - Messina, Italy}
\email{bonanno@@unime.it}
\address[G. Molica Bisci]{Dipartimento MECMAT, University of Reggio Calabria,
 Via Graziella, Feo di Vito, 89124 Reggio Calabria, Italy} \email{gmolica@@unirc.it}
\address[V. R\u{a}dulescu]{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania \& Department of Mathematics, University of Craiova,  Street A.I. Cuza No. 13, 200585 Craiova,
Romania} \email{vicentiu.radulescu@@imar.ro}

\keywords{Sierpi\'nski gasket, Nonlinear elliptic equation; Dirichlet form; Weak Laplacian.}

\thanks{{\em 2010 Mathematics Subject Classification.} Primary 35J20; Secondary 28A80, 35J25, 35J60,
47J30, 49J52}

\newtheorem{theorem}{Theorem}[section]
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\newtheorem{example}{Example}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{th}{Theorem}
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\begin{document}


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\begin{abstract}
Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the
Sierpi\'nski gasket is proved. Our approach is based on
variational methods and on some analytic and geometrical properties of the Sierpi\'nski fractal. The abstract results are illustrated by explicit examples.
\end{abstract}

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\maketitle
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\section{Introduction}
In the last three decades of the 20th century it started to become clear that many phenomena
in the real world are best modeled by some exotic geometric structures with a nonsmooth appearance. The theory of fractals, as Benoit B.~Mandelbrot
\cite{ManUK,Man,Man1} has so forcefully argued, seeks to provide the
mathematical framework for such powerful development.
Analysis of PDEs on fractal domains has shown an explosive development, due to numerous applications to problems arising in various fields, including physics,
chemistry and biology. In this paper we study the following Dirichlet problem
\begin{equation}\tag{$S_{a,\lambda}^{f,g}$}\label{N}
\left\{
\begin{array}{l}
\Delta u(x)+a(x)u(x)=\lambda g(x)f(u(x))\quad x \in V\setminus V_0, \\
u|_{V_0}=0,\\
\end{array}
\right.
\end{equation}
where $V$ stands for the Sierpi\'nski gasket, $V_0$ is its intrinsic boundary, $\Delta$ denotes the weak Laplacian on $V$ and $\lambda$ is a positive real parameter. We assume that $f:\R\to\R$ is a continuous function and that the variable potentials $a,\, g:V\to \R$ satisfy the following conditions:
\begin{itemize}
\item[{$({\rm{h}}_1)$}] $a\in L^1(V,\mu)$ and $a\leq0$ almost everywhere in $V$;
\item[{$({\rm{h}}_2)$}] $g\in C(V)$ with $g\leq0$ and such that the restriction of $g$ to every open subset of $V$ is not identically zero.
\end{itemize}
Here $\mu$ denotes the restriction to $V$ of normalized ${\log N}/{\log 2}$-dimensional Hausdorff measure on $V$, so that $\mu(V)=1$.\par
\indent In our main result just requiring an oscillating behaviour of the
non-linearity $f$ either at zero or at infinity, we prove that problem (\ref{N}) admits a sequence of
pairwise distinct weak solutions; see Theorems \ref{Main1} and
\ref{Main2} below.\par
The Sierpi\'nski gasket has the origin in a paper by Sierpi\'nski \cite{Sie}. In a very simple manner, this fractal domain
can be described as a subset of the plane obtained from an equilateral triangle by removing the open middle inscribed equilateral triangle
of $1/4$ of the area, removing the corresponding open triangle from each of the three constituent triangles and continuing in this way. This fractal can also be obtained as
the closure of the set of vertices arising in this
construction.\par
 Over the years, the Sierpi\'nski gasket showed both to be
extremely useful in representing roughness in nature
and man's works.
This geometrical object
is one of the most familiar examples of fractal domains and
it gives insight into the turbulence of fluids. According to Kigami
\cite{K}, this notion was introduced by Mandelbrot \cite{Man} in
1977 to design  a class of mathematical objects which are not
collections of smooth components. We refer to Strichartz \cite{stri} for an elementary introduction to this subject and to Strichartz \cite{stribook} for important applications to differential equations on fractals.\par

The study of the Laplacian on
fractals was originated in physics literature, where so-called
{\it spectral decimation method} was developed in Alexander \cite{A} and Rammal {\it et al.} \cite{Ra,Ra2}. The
Laplacian on the Sierpi\'nski gasket was first constructed as the
generator of a diffusion process by Kusuoka \cite{Ku} and Goldstein \cite{gold}.
Among the contributions to the theory of nonlinear elliptic
equations on fractals we mention \cite{BRV,Fa99,FaHu} and \cite{Hu,HuaZh,stripaper}.\par
 The main tools used in these papers to prove
the existence of at least one nontrivial solution or of multiple
solutions for nonlinear elliptic equations with zero Dirichlet
boundary conditions are certain mini-max results (mountain pass
theorems, saddle-point theorems), respectively, mi\-ni\-mi\-zation
procedures.\par
In this note, starting from the seminal paper by Falconer and Hu \cite{FaHu}, we study the Dirichlet problem (\ref{N}) through variational methods. In particular, in order to prove our multiplicity result we use the following critical points theorem obtained by Bonanno and Molica Bisci in \cite{BMB} that we recall here in a convenient form; see also the variational principle of Ricceri, contained in \cite{Ricc2}.
\begin{theorem}\cite[Theorem 2.1]{BMB}\label{BMB}
Let $X$ be a reflexive real Banach space, and $\Phi,\Psi:X\to\R$ be
two G\^{a}teaux differentiable functionals such that $\Phi$ is
strongly continuous, sequentially weakly lower semi-continuous and
coercive and $\Psi$ is sequentially weakly upper semi-continuous. For every $r>\inf_X \Phi$, put
$$
\varphi(r):=\inf_{u\in \Phi^{-1}(]-\infty,r[)}\frac{\left(\sup_{v\in \Phi^{-1}(]-\infty,r[)}\Psi(v)\right)-\Psi(u)}{r-\Phi(u)},
$$
and
$$
\gamma:=\liminf_{r\to +\infty}\varphi(r),\qquad\delta:=\di\liminf_{r\to (\inf_X \Phi)^+}\varphi(r).
$$
Then, one has
\begin{itemize}
\item[\rm{(a)}] If $\gamma<+\infty$ then, for each
$\lambda\in]0,{1}/{\gamma}[$, the following
alternative holds:\par
\noindent either\par
\noindent {\rm $(\rm{a}_1)$} $I_\lambda:=\Phi-\lambda\Psi$ possesses a global minimum,\par
\noindent or\par
\noindent {\rm $(\rm{a}_2)$} there is a sequence $\{v_n\}$ of critical points
$($local minima$)$ of $I_\lambda$ such that $\lim_{n\to \infty}
\Phi(v_n)=+\infty$.
\item[\rm{(b)}] If $\delta<+\infty$ then, for each
$\lambda\in]0,{1}/{\delta}[$, the following
alternative holds:\par
\noindent either\par
\noindent {\rm $(\rm{b}_1)$} there is a global minimum of $\Phi$ which is a local minimum of $I_\lambda$,\par
\noindent or\par
\noindent {\rm $(\rm{b}_2)$} there is a sequence $\{v_n\}$ of pairwise distinct
critical points $($local minima$)$ of $I_\lambda$ which weakly
converges to a global minimum of $\Phi$, with $\lim_{n\to \infty}
\Phi(v_n)=\inf_X\Phi$.
\end{itemize}
\end{theorem}
The above theoretical result assures the existence of a sequence of pairwise distinct critical points for G\^{a}teaux differentiable functionals under assumptions that, when we consider the energy functional associated to (\ref{N}), are satisfied assuming an appropriate oscillating behaviour on the potential of the nonlinearity either at infinity or at the origin. In this order of ideas, very recently, Theorem \ref{BMB} has been used to prove the existence of sequences of solutions for different classes of elliptic problems with Dirichlet boundary condition; see, for example, Bonanno and Molica Bisci in \cite{BMB2}; Bonanno, Molica Bisci and O'Regan in \cite{BMB3} and D'Agu\`{i} and  Molica Bisci \cite{DMB}.\par
  In the cited papers, for technical reasons, a necessary condition applying our abstract result is the presence of suitable classes of test functions. For instance, one admissible set of maps was introduced by  Bonanno and Livrea in \cite{BL} and successfully used in several works. It is worth noticing that the different test functions introduced here are fundamental proving Theorems \ref{Main1} and \ref{Main2} since the usual cannot be used in elliptic problems on Sierpi\'nski gasket due to the particular geometry of the fractal set.\par
   Moreover, we just mention that most results for classical Dirichlet problems on a bounded domain $\Omega\subset \R^N$ assume that the datum $f$ is odd in order to apply some variant of the classical
Lusternik--Schnirelmann theory. Only a few papers deal with non-linearities
having no symmetry properties; see, for instance, the papers of Omari and Zanolin \cite{OZ,OZ0} and the recent work of Krist\'{a}ly and Moro\c{s}anu \cite{KM} for perturbed Dirichlet equations. In analogy with the contributions obtained in \cite{BMB2,BMB3, DaMo1}
in our approach here we don't require any symmetry hypothesis.\par
Finally, we recall that Breckner, R\u{a}dulescu and Varga \cite{BRaduV} proved the existence of infinitely many solutions of problem (\ref{N}) under the key assumption, among others, that the non-linearity $f$ is non-positive in a sequence of positive intervals. We point out that our results are mutually independent compared to those achieved in the above mentioned manuscript; see Remark \ref{Comparition}.

 This paper is organized as follows. In Section 2 we recall the geometrical construction of the Sierpi\'nski gasket and our variational framework. Successively, Section 3 is devoted to the main theorem and finally, in the last section, we give two applications of the presented results. We cite the very recent monograph by Krist\'aly, R\u adulescu and Varga \cite{KRV} as general reference for the basic notions used here.

\section{Abstract Framework}
Let $N\geq2$ be a natural number and let $p_1,\dots, p_N\in\erre^{N-1}$
be so that $|p_i-p_j|=1$ for $i\neq j$. Define, for every
$i\in\{1,\dots,N\}$, the map $S_i\colon\R^{N-1}\to\R^{N-1}$ by
$$S_i(x)=\frac12\,x+\frac12\,p_i\,.$$
Let
${\mathcal S}:=\{S_1,\dots, S_N\}$ and denote by $G\colon{\mathcal
P}(\R^{N-1})\to{\mathcal P}(\R^{N-1})$ the map assigning to a subset
$A$ of $\R^{N-1}$ the set
$$G(A)=\bigcup_{i=1}^NS_i(A).$$
It is known that there
is a unique non-empty compact subset $V$ of $\R^{N-1}$, called the
attractor of the family ${\mathcal S}$, such that $G(V)=V$; see, Theorem 9.1 in Falconer \cite{Fa}.\par
 The set $V$ is called
the \textit{Sierpi\'nski gasket} in $\R^{N-1}$. It can be
constructed inductively as follows:\par
 Put $V_0:=\{p_1,\dots,p_N\}$,
$V_m:=G(V_{m-1})$, for $m\geq1$, and $V_*:=\bigcup_{m\geq0}V_m$. Since
$p_i=S_i(p_i)$ for $i\in\{1,...,N\}$, we have $V_0\subseteq V_1$,
hence $G(V_*)=V_*$. Taking into account that the maps $S_i$,
$i\in\{1,...,N\}$, are homeomorphisms, we conclude that
$\overline{V_*}$ is a fixed point of $G$. On the other hand,
denoting by $C$ the convex hull of the set $\{p_1,\dots, p_N\}$, we
observe that $S_i(C)\subseteq C$ for $i={1,...,N}$. Thus
$V_m\subseteq C$ for every $m\in\N$, so $\overline{V_*}\subseteq C$.
It follows that $\overline{V_*}$ is non-empty and compact, hence
$V=\overline{V_*}$.\par
 In the sequel $V$ is considered to be endowed
with the relative topology induced from the Euclidean topology on
$\R^{N-1}$. The set $V_0$ is called the {\textit{intrinsic
boundary}} of $V$. By Theorem 9.3 in Falconer
\cite{Fa}, the Hausdorff (fractal) dimension $d$ of $V$ satisfies the equality
$$\sum_{i=1}^N\left(\frac12\right)^d=1,$$
hence $d=\displaystyle{\ln N}/{\ln 2}$, and $0<{\mathcal H}^d(V)<\infty$,
where ${\mathcal H}^d$ is the $d$-dimensional Hausdorff measure on
$\R^{N-1}$. We point out that the Hausdorff dimension of a set is a more refined notion than the topological
dimension. In particular, the Hausdorff dimension is closely related to entropy. To see this, we consider the
map $T(z) = z^2$ defined on the circle. If $\nu$ is any ergodic $T$--invariant measure,
then the Kolmogorov entropy of $T$ is (up to a
constant depending on the base of the logarithm
in the definition of entropy) the infimum of the
Hausdorff dimension of subsets of the circle that
have full $\nu$ measure. In the other direction, if $E$ is
a closed subset of the circle that is invariant under
the map $T$, then $E$ supports an invariant measure
whose entropy is (again up to that constant) the
Hausdorff dimension of the set $E$.

Let $\mu$ be the normalized restriction of ${\mathcal
H}^d$ to the subsets of $V$, so $\mu(V)=1$. Finally, the following property
of $\mu$ will be useful in the sequel:
\begin{equation}\label{support}
\mu(B)>0,\hbox{ for every non-empty open subset $B$ of }V.
\end{equation}
In other words, the support of $\mu$ coincides with $V$; see, for instance, Breckner, R\u adulescu and Varga \cite{BRaduV}.\par
\indent Denote by
$C(V)$ the space of real-valued continuous functions on $V$ and by
$$C_0(V):=\{u\in C(V)\mid u|_{V_0}=0\}.$$
The spaces $C(V)$ and $C_0(V)$ are endowed with the usual supremum
norm $\|\cdot\|_\infty$. For a function $u\colon V\to\R$ and for $m\in\N$
let
\begin{equation}\label{defWm}
W_m(u)=\left(\frac{N+2}{N}\right)^m\sum_{\underset{|x-y|=2^{-m}}{x,y\in
V_m}}(u(x)-u(y))^2.
\end{equation}
We have $W_m(u)\leq W_{m+1}(u)$ for every natural $m$, so we can put
\begin{equation}\label{defW}
W(u)=\lim_{m\to\infty} W_m(u).
\end{equation}
Define now
$$H_0^1(V):=\{u\in C_0(V)\mid W(u)<\infty\}.$$
It turns out that $H_0^1(V)$ is a dense linear subset of
$L^2(V,\mu)$ equipped with the $\|\cdot\|_2$ norm. We now endow
$H_0^1(V)$ with the norm
$$\|u\|=\sqrt{W(u)}.$$
In fact, there is an inner product defining this norm: for $u,v\in
H_0^1(V)$ and $m\in\N$ let
$${\mathcal W}_m(u,v)=\left(\frac{N+2}{N}\right)^m\sum_{\underset{|x-y|=2^{-m}}{x,y\in V_m}}(u(x)-u(y))(v(x)-v(y)).$$
Put
$${\mathcal W}(u,v)=\lim_{m\to\infty} {\mathcal W}_m(u,v).$$

\indent Then ${\mathcal W}(u,v)\in\R$ and the space $H_0^1(V)$, equipped with the
inner product ${\mathcal W}$, which induces the norm $\|\cdot\|$,
becomes a real Hilbert space.\par \noindent Moreover,
\begin{equation}\label{embeddingconstant}
\|u\|_\infty\leq(2N+3)\|u\|, \hbox{ for every }u\in H_0^1(V),
\end{equation}
and the embedding
\begin{equation}\label{embedding}
(H_0^1(V),\|\cdot\|)\hookrightarrow (C_0(V),\|\cdot\|_\infty)
\end{equation}
is compact. We refer to Fukushima and Shima \cite{FuSc} for further details.

\begin{remark}\label{equivalent}
\rm{
As pointed out by Falconer and Hu \cite{FaHu}, we observe that if $a\in L^1(V)$ and $a\leq 0$ in $V$ then, from (\ref{embeddingconstant}),
the norm
$$
\|u\|_*:=\left({\mathcal W}(u,u)-\int_Va(x)u^2d\mu\right)^{1/2},
$$
is equivalent to $\sqrt{W(u)}$ in $H^1_0(V)$.
}
\end{remark}

\indent We now state a useful property of the space $H_0^1(V)$ which shows,
together with the facts that $(H_0^1(V),\|\cdot\|)$ is a Hilbert
space and  $H_0^1(V)$ is dense in $L^2(V,\mu)$, that ${\mathcal
W}$ is a Dirichlet form on $L^2(V,\mu)$.

\begin{lemma}\label{MaMianalogue}
Let $h\colon \R\to\R$ be a Lipschitz mapping with Lipschitz constant
$L\geq0$ and such that $h(0)=0$. Then, for every $u\in H_0^1(V)$, we
have $h\circ u\in H_0^1(V)$ and $\|h\circ u\|\leq L\|u\|$.
\end{lemma}
\begin{pf}
It is clear that $h\circ u\in C_0(V)$. For every $m\in\N$ we have,
by (\ref{defWm}) and the Lipschitz property of $h$, that
$$W_m(h\circ u)\leq L^2 W_m(u).$$
Hence $W(h\circ u)\leq L^2 W(u)$, according to (\ref{defW}). Thus
$h\circ u\in H_0^1(V)$ and $\|h\circ u\|\leq L\|u\|$.
\end{pf}

\indent Following Falconer and Hu \cite{FaHu}
we can define in a standard way a linear self-adjoint operator $\Delta\colon Z\to L^2(V,\mu)$,
where $Z$ is a linear subset of $H_0^1(V)$ which is dense in $L^2(V,\mu)$ (and dense also in $(H_0^1(V),\|\cdot\|)$),
such that
$$-{\mathcal W}(u,v)=\int_V\Delta u\cdot vd\mu,\hbox{ for every }(u,v)\in Z\times H_0^1(V).$$
The operator $\Delta$ is called the {\textit{(weak) Laplacian} on}
$V$.\par \noindent Precisely, let $H^{-1}(V)$ be the closure of
$L^2(V,\mu)$ with respect to the pre-norm
$$
\|u\|_{-1}=\sup_{\underset{\|h\|=1}{h\in H^1_0(V)}} |<u,h>|,
$$
\noindent where
$$
<v,h>=\int_Vvhd\mu,
$$
\noindent $v\in L^2(V,\mu)$ and $h\in H^1_0(V)$. Then $H^{-1}(V)$ is a Hilbert space. Then the relation
$$
-{\mathcal W}(u,v)=<\Delta u,v>,\,\,\,\,\,\,\forall v\in H^1_0(V),
$$
uniquely defines a function $\Delta u\in H^{-1}(V)$ for every $u\in
H^1_0(V)$.\par \noindent Finally, fix $\lambda>0$. Given $a\colon
V\to\R$, $f\colon\R\to\R$ and $g\colon V\to\R$ be as in
Introduction. We say that a function $u\in H_0^1(V)$ is called a
{\textit{weak solution} of} (\ref{N}) if
$${\mathcal W}(u,v)-\int_Va(x)u(x)v(x)d\mu+\lambda\int_Vg(x)f(u(x))v(x)d\mu=0,$$
for every $v\in H_0^1(V)$.\par

While we mainly work with the weak Laplacian, there is also a directly
defined version. We say that $\Delta_s$ is the \textit{standard Laplacian} of $u$ if $\Delta_su:V\to\R$ is continuous and
$$
\lim_{m\to\infty}\sup_{x\in V\setminus V_0}|(N+2)^m(H_mu)(x)-\Delta_su(x)|=0,
$$
\noindent where
$$
(H_mu)(x):=\sum_{\underset{|x-y|=2^{-m}}{y\in V_m}}(u(y)-u(x)),
$$
\noindent for $x\in V_m$. We say that $u\in C_0(V)$ is a {\it strong solution} of (\ref{N}) if $\Delta_su$ exists and is continuous for all $x\in V\setminus V_0,$ and
$$
\Delta u(x)+a(x)u(x)=\lambda g(x)f(u(x)),\,\,\,\,\,\forall\; x\in V\setminus V_0.
$$
The existence of the standard Laplacian of a function $u\in H^1_0(V)$ implies
the existence of the weak Laplacian $\Delta$; see, for completeness, Falconer and Hu \cite{FaHu}.\par
\begin{remark}\label{strong}
\rm{
If $a\in C(V)$, $f\colon\R\to\R$ is continuous and $g\in C(V)$,
then,  using  the regularity result Lemma 2.16 of Falconer and Hu \cite{FaHu}, it
follows that every weak solution of the problem (\ref{N}) is also a
strong solution.}
\end{remark}
\section{Main results}

 Define $F\colon\R\to\R$ by $F(\xi)=\displaystyle\int_0^\xi f(t)dt$, for every $\xi\in\R$, and fix $\lambda>0$.
The functional $I\colon H_0^1(V)\to\R$  given by
\begin{equation}\label{energyfunctional}
I_\lambda(u)=\frac12\|u\|^2-\frac12\int_Va(x)u^2(x)d\mu+\lambda\int_Vg(x)F(u(x))d\mu,\
\forall\; u\in H_0^1(V),
\end{equation}
will turns out to be the the energy functional attached to problem
(\ref{N}). To see this we first recall, for completeness, a few
basic notions.

\begin{definition}
Let $(E,\|\cdot\|)$ be a real Banach space, $E^*$ its topological dual and
$T\colon E\to\R$ a functional. We say that $T$ is {\it Fr\'{e}chet
differentiable at} $u\in E$ if there exists a continuous linear map
$T'(u)\colon E\to\R$, called the {\it Fr\'{e}chet differential of
$T$ at $u$}, such that
$$\lim_{v\to 0}\frac{|T(u+v)-T(u)-T'(u)(v)|}{\|v\|}=0.$$
The functional $T$ is {\it Fr\'{e}chet differentiable on $E$} if $T$
is Fr\'{e}chet differentiable at every point $u\in E$. A point $u\in
E$ is a {\it critical point of} $T$ if $T$ is Fr\'{e}chet
differentiable at $u$ and if $T'(u)=0$. Moreover, if $T':E\rightarrow
E^*$ is continuous, then T is called a $C^1(E,\R)$ functional.
\end{definition}

\begin{remark}\label{useful}
Note that if the functional $T\colon E\to\R$ has in $u\in E$ a local
extremum  and if $T$ is Fr\'{e}chet differentiable at $u$, then $u$
is a critical point of $T$.
\end{remark}

We have the following result contained in \cite[Proposition 2.19]{FaHu} that we recall here in a convenient form.
\begin{lemma}\label{criticalpoint}
The functional $I_\lambda\colon H_0^1(V)\to\R$ defined by relation
{\rm (\ref{energyfunctional})} is a $C^1(H_0^1(V),\R)$ functional.
Moreover,
$$I_\lambda'(u)(v)={\mathcal W}(u,v)-\int_Va(x)u(x)v(x)d\mu+\lambda\int_Vg(x)f(u(x))v(x)d\mu,\,\,\,\forall\; v\in H^1_0(V)$$
for each point $u\in H^1_0(V)$. In particular, $u\in H_0^1(V)$ is a weak solution of problem (\ref{N})
if and only if $u$ is a critical point of $I_\lambda$.
\end{lemma}


\indent The aim of the paper is to prove the following result concerning the
existence of infinitely many weak solutions of the problem (\ref{N}).

\begin{theorem}\label{Main1}
Let $f:\R\rightarrow\R$ be a non-negative continuous function.
Assume that
$$
\displaystyle \liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{2}}<+\infty\,\,\,\,\,\,\,\ \textrm{and}\,\,\,\,\,\,\,\,\,\limsup_{\xi\rightarrow 0^{+}
}\frac{F(\xi)}{\xi^{2}}=+\infty.\eqno{(\rm{h}_{0})}
$$
\noindent Then, for every
$$
\lambda\in\left]0,-\frac{1}{2(2N+3)^2\left(\di\int_Vg(x)d\mu\right) \displaystyle \liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{2}}}\right[,
$$
there exists a sequence $\{v_n\}$ of pairwise distinct weak solutions
of problem \eqref{N} such that
$\displaystyle\lim_{n\to\infty}\|v_n\|=\displaystyle\lim_{n\to\infty}\|v_n\|_\infty=0.$
\end{theorem}

\begin{pf}
Let us define the functionals
$\Phi,\Psi:X\rightarrow\RR$ by
$$\Phi(u)=\frac12\|u\|^2-\frac12\int_Va(x)u^2(x)d\mu \ \ \ \mbox{and}\ \ \ \Psi(u)=-\int_Vg(x)F(u(x))d\mu,$$
 where $X$ denotes the reflexive Banach space $H^1_0(V)$. Now, in order to achieve our goal, fix $\lambda$ as in the conclusion. Clearly, with the above notations, one has that $I_\lambda=\Phi-\lambda\Psi$.
Hence, we seek for
weak solutions of problem (\ref{N}) by applying part (\rm{b}) of Theorem \ref{BMB}. First of all we observe that, from Lemma \ref{criticalpoint}, the functional $I_\lambda\in C^{1}(X,\R)$. Moreover, $\Phi$ is obviously coercive and, by using Lemma 5.6 in Breckner, R\u adulescu and Varga \cite{BRaduV}, the functionals $\Phi$ and $\Psi$ are weakly sequentially lower
semi-continuous on $X$. \noindent Now, let $\{c_n\}$ be a real sequence such
that $\lim_{n\rightarrow \infty}c_n=0$ and
$$
\lim_{n\rightarrow \infty}\frac{F(c_n)}{c_n^{2}}=\displaystyle \liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{2}}.
$$
\noindent Put $\displaystyle r_n=\frac{c_n^2}{2(2N+3)^2}$ for every $ n \in \N$.
Due to the compact embedding into $C_0(V)$, from (\ref{embeddingconstant}), we have
$$
\left\{v\in X \mid \Phi(v)<r_n \right\}\subseteq \left\{v\in X \mid \|v\|_\infty\leq c_n \right\}.
$$
Therefore
$$
\varphi(r_n)=\inf_{\Phi(u)<r_n}\frac{\displaystyle\sup_{\Phi(v)<
r_n}\displaystyle\int_V(-g(x))F(v(x))d\mu+\int_Vg(x)F(u(x))d\mu}{r_n-\Phi(u)}
$$
$$
\leq\frac{\displaystyle\sup_{\Phi(v)<
r_n}\displaystyle\int_V(-g(x))F(v(x))d\mu}{r_n}\leq -\left(\int_Vg(x)d\mu\right)\frac{\displaystyle\max_{|\xi|\leq c_n}F(\xi)}{r_n}
$$
$$
=
-\left(\di\int_Vg(x)d\mu\right)\frac{F(c_n)}{r_n}=-2(2N+3)^2\left(\di\int_Vg(x)d\mu\right)\frac{F(c_n)}{c_n^{2}}.\,\,\,\,\,\,\, %\forall n\in \enne.
$$
Thus, since
$$
\displaystyle \liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{2}}<+\infty,
$$
we deduce that
$$
\displaystyle \delta\leq\liminf_{n\rightarrow
\infty}\varphi(r_n)\leq -2(2N+3)^2\left(\di\int_Vg(x)d\mu\right) \displaystyle \liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{2}}<+\infty.
$$
\noindent At this point we will show that $0$, that is the unique global minimum of $\Phi$, is not a local minimum of the functional $I_\lambda$. Hence, fix a function $u\in X$ such that there is an element $x_0\in
V$ with $u(x_0)>1$. It follows that
 $$
 D:=\{x\in V\mid u(x)>1\}
 $$
 is a non-empty open (from the continuity of $u$) subset of $V$. Define $h\colon\R\to\R$ as follows
$$
h(t)=|\min\{t,1\}|,\qquad\mbox{for all}\ t\in\R.
$$
Then $h(0)=0$ and $h$ is a Lipschitz function whose Lipschitz constant $L$ is equal to $1$. Hence, by using Lemma
\ref{MaMianalogue}, it follows that $v:=h\circ u\in X$. Moreover,
$v(x)=1$ for every $x\in D$, and $0\leq v(x)\leq1$ for every $x\in
V$.
Bearing in mind that
$$
\limsup_{\xi\rightarrow 0^{+}
}\frac{F(\xi)}{\xi^{2}}=+\infty,
$$
 there exists a sequence $\{\xi_n\}$
in $]0,\rho[$ such that $\displaystyle\lim_{n\to\infty}\xi_n=0$ and
\begin{equation}\label{infinito}
\lim_{n\to\infty}\frac{F(\xi_n)}{\xi_n^2}=+\infty.
\end{equation}
Consider the sequence of functions $\{\xi_nv\}\subset X$. Clearly $\|\xi_nv\|\rightarrow 0$ and
\begin{eqnarray*}
I_\lambda(\xi_nv)=\frac{\xi_n^2}{2}\|v\|^2-\frac{\xi_n^2}{2}\int_Va(x)v^2(x)d\mu&+&\lambda F(\xi_n)\int_Dg(x)d\mu\\
&+&\lambda\int_{V\setminus D}g(x)F(\xi_nv(x))d\mu.
\end{eqnarray*}
\noindent Taking into account that $F$ is positive in $]0,+\infty[$, from hypothesis $({\rm{h}}_2)$, the above equation becomes
\begin{eqnarray*}
I_\lambda(\xi_nv)\leq\frac{\xi_n^2}{2}\|v\|^2-\frac{\xi_n^2}{2}\int_Va(x)v^2(x)d\mu+\lambda F(\xi_n)\int_Dg(x)d\mu,\,\,\,\,\,\,\forall\; n\in \N.
\end{eqnarray*}
\noindent Thus
$$\frac{I_\lambda (\xi_nv)}{\xi_n^2}\leq \frac12\|v\|^2-\frac12\int_Va(x)v^2(x)d\mu+\lambda\frac{F(\xi_n)}{\xi_n^2}\int_Dg(x)d\mu,$$
\noindent for every $n\in\N$.\par
\noindent Moreover, conditions $({\rm{h}}_2)$ and (\ref{support}) imply that $\di\int_Dg(x)d\mu<0$.
Now, thanks to (\ref{infinito}),
 the above computations ensure that
$$\lim_{n\to\infty}\frac{I_\lambda(\xi_nv)}{\xi_n^2}=-\infty.$$
 \noindent Thus $I_\lambda(\xi_nv)<0$ for every $n$ sufficiently large. Since $I_\lambda(0_X)=\Phi(0_X)-\lambda \Psi(0_X)=0$, this means that $0_X$ is not a local minimum of $I_\lambda$. Moreover, since $\Phi$ has $0_X$ as unique global minimum, Theorem \ref{BMB} ensures the existence of a sequence $\{v_n\}$ of pairwise distinct critical points of the functional $I_\lambda$, such that
  $$\lim_{n\rightarrow\infty}\left(\|v_n\|^2-\int_Va(x)v_n^2(x)d\mu\right)=0.$$
\noindent Hence, one has that $\displaystyle\lim_{n\rightarrow \infty}\|v_n\|=0$. In particular, by (\ref{embedding}), it follows that
$\displaystyle\lim_{n\to\infty}\|v_n\|_\infty=0$. Then, the proof is complete bearing in mind Lemma \ref{criticalpoint}.
\end{pf}

By the same method, applying part (\rm{a}) instead of part \rm{(b)}
of Theorem \ref{BMB}, one can prove the following analogous result
in the case when the nonlinear term $f\colon\R\to\R$ has an oscillating
behaviour at infinity. In this setting one obtains a sequence
$\{v_n\}$ of weak solutions to problem (\ref{N})
such that $\displaystyle\lim_{n\to\infty}\|v_n\|=+\infty$.


\begin{theorem}\label{Main2}
Let $f:\R\rightarrow\R$ be a non-negative continuous function.
Assume that
$$
\displaystyle \liminf_{\xi\rightarrow +\infty}\frac{F(\xi)}{\xi^{2}}<+\infty\,\,\,\,\,\,\,\ \textrm{and}\,\,\,\,\,\,\,\,\,\limsup_{\xi\rightarrow +\infty
}\frac{F(\xi)}{\xi^{2}}=+\infty.\eqno{(\rm{h}_{\infty})}
$$
\noindent Then, for every
$$
\lambda\in\left]0,-\frac{1}{2(2N+3)^2\left(\di\int_Vg(x)d\mu\right) \displaystyle \liminf_{\xi\rightarrow +\infty}\frac{F(\xi)}{\xi^{2}}}\right[,
$$
problem \eqref{N} admits a sequence of weak solutions which is
unbounded in $H^1_0(V)$.
\end{theorem}
\begin{pf}
The strategy of the proof is very similar to the previous one. Hence, in the sequel, we omit the details and we use the notations adopted showing  Theorem \ref{Main1}. Then, from hypothesis
$$
\displaystyle \liminf_{\xi\rightarrow +\infty}\frac{F(\xi)}{\xi^{2}}<+\infty,
$$
\noindent by direct computations, it follows that $\gamma:=\di\liminf_{r\to +\infty}\varphi(r)<+\infty$. On the other hand by
$$
\limsup_{\xi\rightarrow +\infty
}\frac{F(\xi)}{\xi^{2}}=+\infty,
$$
there exists a sequence $\{\eta_n\}$ of positive constants such that $\displaystyle\lim_{n\to\infty}\eta_n=+\infty$ and
\begin{equation}\label{mic52}
\lim_{n\to\infty}\frac{F(\eta_n)}{\eta_n^2}=+\infty.
\end{equation}

\noindent Now, consider the sequence of functions $\{\eta_nv\}\subset X$. Arguing as in Theorem \ref{Main1}, we obtain
$$\lim_{n\to\infty}\frac{I_\lambda(\eta_nv)}{\eta_n^2}=-\infty.$$
  Hence, the functional $I_\lambda$ is unbounded from below. Then, part \rm{(a)} of Theorem \ref{BMB} ensures the existence of a sequence $\{v_n\}$ of distinct critical points of $I_\lambda$ such that $$\lim_{n\rightarrow\infty}\left(\|v_n\|^2-\int_Va(x)v_n^2(x)d\mu\right)=+\infty.$$ In conclusion, by Remark \ref{equivalent}, it follows that $\displaystyle\lim_{n\rightarrow \infty}\|v_n\|=+\infty$. This completes the proof.
\end{pf}
\begin{remark}\rm{
If to the hypotheses of Theorem \ref{Main1}, respectively in Theorem \ref{Main2}, one adds the
requirement that $a\in C(V)$, then our results and Remark
\ref{strong} yield the existence of a sequence of pairwise distinct
strong solutions of problem (\ref{N}) that either converges to zero or is unbounded in $H^1_0(V)$.}
\end{remark}

\begin{remark}\label{general}\rm{

We explicitly observe that, exploiting the proof of Theorem
\ref{Main1}, one can see that our result also holds for
sign-changing functions $f:\R\rightarrow \R$ just requiring that
$$
-\infty<\liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{2}},\,\,\,\,\,\,
\displaystyle \liminf_{\xi\rightarrow 0^+}\frac{{\max_{t \in [-\xi, \xi]}F(t)}}{\xi^{2}}<+\infty\,\,\,\,\,\ \textrm{and}\,\,\,\,\,\,\,\,\limsup_{\xi\rightarrow 0^+
}\frac{F(\xi)}{\xi^{2}}=+\infty,
$$
\noindent instead of condition $(\rm{h}_{0})$. In this setting, for every
$$
\lambda\in\left]0,-\frac{1}{2(2N+3)^2\left(\di\int_Vg(x)d\mu\right) \displaystyle \liminf_{\xi\rightarrow 0^+}\frac{{\max_{t \in [-\xi, \xi]}F(t)}}{\xi^{2}}}\right[,
$$
there exists a sequence $\{v_n\}$ of pairwise distinct weak solutions
of problem (\ref{N}) such that
$\displaystyle\lim_{n\to\infty}\|v_n\|=\displaystyle\lim_{n\to\infty}\|v_n\|_\infty=0.$\par
\indent An analogous conclusion can be achieved if the potential $F$ has the same behaviour at infinity instead of at zero obtaining, in this case, the existence of a sequence of weak solutions which is
unbounded in $H^1_0(V)$. Indeed, with the notations of Theorem \ref{Main2}, if
$$
\liminf_{\xi\rightarrow +\infty}\frac{{\max_{t \in [-\xi, \xi]}F(t)}}{\xi^{2}}<+\infty,
$$
direct computations ensure that $\gamma<+\infty.$
On the other hand, consider the sequence of functions $\{\eta_nv\}\subset H^1_0(V)$. Bearing in mind that $\di\liminf_{\xi\rightarrow +\infty
}\frac{F(\xi)}{\xi^{2}}>-\infty$, there exist $\varrho>0$ and a real constant $k$ such that
\begin{equation}\label{mic5}
F(\xi)\geq k\xi^2,\,\hbox{ for every } \xi\in ]\varrho,+\infty[.
\end{equation}
Moreover, one has
\begin{eqnarray*}
I_\lambda(\eta_nv)=\frac{\eta_n^2}{2}\|v\|^2-\frac{\eta_n^2}{2}\int_Va(x)v^2(x)d\mu&+&\lambda F(\eta_n)\int_Dg(x)d\mu\\
&+&\lambda\int_{V\setminus D}g(x)F(\eta_nv(x))d\mu,
\end{eqnarray*}
\noindent for every $n\in \enne$.\par
\noindent Hence
\begin{eqnarray*}
I_\lambda(\eta_nv)=\frac{\eta_n^2}{2}\|v\|^2-\frac{\eta_n^2}{2}\int_Va(x)v^2(x)d\mu&+&\lambda F(\eta_n)\int_Dg(x)d\mu\\
&+&\lambda\int_{G_{\varrho}\cap(V\setminus D)}g(x)F(\eta_nv(x))d\mu\\
&+&\lambda\int_{G^{\varrho}\cap(V\setminus D)}g(x)F(\eta_nv(x))d\mu,
\end{eqnarray*}
\noindent where
$$
G_\varrho:=\{x\in V:0\leq \eta_nv(x)\leq\varrho\}\,\,\,\,\,\,\,\,{\rm and}\,\,\,\,\,\,\,\,G^{\varrho}:=\{x\in V:\eta_nv(x)>\varrho\}.
$$
\noindent Now, by using the mean value theorem, it easy to see that
\begin{equation}\label{glo}
\int_{G_{\varrho}\cap(V\setminus D)}g(x)F(\eta_nv(x))d\mu\leq \|g\|_\infty\max_{t\in [0,\varrho]}|f(t)|\varrho.
 \end{equation}
 Then, inequalities (\ref{mic5}) and (\ref{glo}) yield
\begin{eqnarray*}
I_\lambda(\eta_nv)\leq \frac{\eta_n^2}{2}\|v\|^2-\frac{\eta_n^2}{2}\int_Va(x)v^2(x)d\mu&+&\lambda F(\eta_n)\int_Dg(x)d\mu\\
&+&\lambda\|g\|_\infty\max_{t\in [0,\varrho]}|f(t)|\varrho\\
&+&\lambda k\eta_n^2\int_{V\setminus D}g(x)v^2(x)d\mu,
\end{eqnarray*}
\noindent for every $n\in\N$.\par
\noindent Thus, condition $({\rm{h}}_2)$ and (\ref{support}) imply that $\di\int_Dg(x)d\mu<0$.
Finally, from (\ref{mic52}) and the above inequality, we have that
$$\lim_{n\to\infty}\frac{I_\lambda(\eta_nv)}{\eta_n^2}=-\infty.$$
Thus $I_\lambda$ is unbounded from below. The proof is attained from part \rm{(a)} of our theoretical result. In conclusion, for every
$$
\lambda\in\left]0,-\frac{1}{2(2N+3)^2\left(\di\int_Vg(x)d\mu\right) \displaystyle \liminf_{\xi\rightarrow +\infty}\frac{{\max_{t \in [-\xi, \xi]}F(t)}}{\xi^{2}}}\right[,
$$
there exists a sequence $\{v_n\}$ of weak solutions
of problem (\ref{N}) which is
unbounded in $H^1_0(V)$.
}
\end{remark}


\section{Examples}

\indent Following Omari and Zanolin in \cite{OZ}, we give a
concrete example of positive continuous function
$q:\R\rightarrow\R$ such that its potential $Q$ satisfies our
growth conditions at zero. Precisely, let $\{s_n\}$, $\{t_n\}$ and
$\{\delta_n\}$ be real sequences defined by
$$
\di s_n:=2^{-\frac{n!}{2}},\,\,\,\,\,\,\,\,\ t_n:=2^{-2n!},\,\,\,\,\,\,\,\, \delta_n:=2^{-(n!)^2}.
$$
\noindent Observe that, definitively, one has
$$
s_{n+1}<t_n<s_n-\delta_n.
$$
\begin{example}\label{Ex.1}
\rm{ \noindent With the above notations, let
$q:\R\rightarrow\R$ be a continuous nondecreasing function
such that $q(t)=0$ in $]-\infty,0]$, $q(t)>0$ for every $t>0$ and
$$
q(t):=2^{-n!},\,\,\,\,\,\,\,\forall\; t\in [s_{n+1},s_{n}-\delta_n],
$$
for $n$ sufficiently large. Define $Q:\R\to\R$ given by
$Q(\xi)=\displaystyle\int_0^\xi q(t)dt$, for every $\xi\in\erre$.
Then
$$
\frac{Q(s_n)}{s_n^2}\leq \frac{q(s_{n+1})s_n+q(s_n)\delta_n}{s_n^2}\rightarrow 0,
$$
\noindent and
$$
\frac{Q(t_n)}{t_n^{2}}\geq \frac{q(s_{n+1})(t_n-s_{n+1})}{t_n^{2}}\rightarrow +\infty.
$$
\noindent Hence
$$
\displaystyle \liminf_{\xi\rightarrow 0^+}\frac{Q(\xi)}{\xi^{2}}=0,\,\,\,\,\,\,\,\ \textrm{and}\,\,\,\,\,\,\,\,\limsup_{\xi\rightarrow 0^{+}
}\frac{Q(\xi)}{\xi^{2}}=+\infty.
$$
\noindent Thus, for every $a,g:V\to\R$ satisfying the conditions $({\rm{h}}_1)$ and $({\rm{h}}_2)$ the following Dirichlet problem
\begin{equation}\tag{$S_{a,\lambda}^{q,g}$}\label{N2}
\left\{
\begin{array}{l}
\Delta u(x)+a(x)u(x)=\lambda g(x)q(u(x))\quad x \in V\setminus V_0, \\
u|_{V_0}=0,\\
\end{array}
\right.
\end{equation}
for every $\lambda>0$, admits a sequence $\{v_n\}$ of pairwise
distinct weak solutions such that
$\displaystyle\lim_{n\to\infty}\|v_n\|=\displaystyle\lim_{n\to\infty}\|v_n\|_\infty=0.$}
\end{example}
\begin{remark}\label{Comparition}
\rm{ In \cite{BRaduV} the authors proved the existence of infinitely
many solutions for problem $(S_{a,1}^{f,g})$. Their result is achieved
under suitable assumptions on the behaviour of the potential $F$ at
zero and requiring that there exist two real positive sequences
$\{a_n\}$ and $\{b_n\}$ with $b_{n+1}<a_n<b_{n}$,
$\displaystyle\lim_{n\to \infty}b_n=0$ and such that $f(t)\leq 0
\hbox{ for every }t\in [a_n,b_n].$ Clearly, the mentioned result
cannot be applied to the cases in which the nonlinearity is strictly
positive in $]0,+\infty[$.}
\end{remark}

Finally, the next example deals with a sign-changing function
$h:\R\rightarrow \R$. The result is achieved taking into account a more
general condition of $(\rm{h}_{\infty})$ in Theorem \ref{Main2} as
indicated in Remark \ref{general}.

\begin{example}\label{Ex.2}
\rm{
\noindent Set
$$a_1:=2,
\,\,\,\,\,\,\,\,\,\,\, a_{n+1}:=(a_n)^{3 \over 2},$$ \noindent for
every $n\in\N$ and $S:=\bigcup_{n\geq 0}]a_{n+1}-1,a_{n+1}+1[$. Define the continuous function
$h:\R\rightarrow\R$ as follows

$$
h(t):=\left\{
\begin{array}{l}
(a_{n+1})^3 e^{{1 \over {(t-(a_{n+1}-1)) (t-(a_{n+1}+1))}}+ 1}{2(a_{n+1}-t) \over {(t-(a_{n+1}-1))^2 (t-(a_{n+1}+1))^2}}   \,\,\,\, \textrm{if}\,\,\, \displaystyle t \in S \\
[1mm]
0 \qquad \qquad \qquad\qquad\qquad \qquad\qquad \qquad \qquad\qquad\qquad\quad{\rm otherwise.}
\end{array}
\right.
$$
\noindent Then
$$
H(\xi)=\int_0^{\xi}h(t)dt=\left\{
\begin{array}{l}
(a_{n+1})^3 e^{{1 \over {(\xi-(a_{n+1}-1)) (\xi-(a_{n+1}+1))}}+ 1}   \,\,\,\, \textrm{if}\,\,\, \displaystyle \xi\in S \\
0 \qquad \qquad \qquad\qquad \qquad \qquad\quad{\rm otherwise,}
\end{array}
\right.
$$

\noindent and $H(a_{n+1})=(a_{n+1})^3$ for every $n \in \N$. Hence, one has
$$
\displaystyle\limsup_{\xi\rightarrow +\infty}\frac{H(\xi)}{\xi^2}=+\infty.
$$
On the other hand, by choosing $x_n=a_{n+1}-1$ for every $n \in \N$, one has $\displaystyle\max_{\xi \in [-x_n, x_n]}H(\xi)=(a_n)^3$ for every $n \in \N$. Moreover
 $$
 \displaystyle\lim_{n \to \infty}{{\max_{\xi \in [-x_n, x_n]}H(\xi)}\over {x_n}^2}=1
 $$
 and, by a direct computation, it follows that
$$
\displaystyle\liminf_{\xi\rightarrow+\infty}{{\max_{t \in [-\xi, \xi]}H(t)}\over {\xi}^2}=1.$$
Hence,
$$
0\leq \liminf_{\xi\rightarrow+\infty}\frac{{H(\xi)}}{\xi^2}\leq\liminf_{\xi\rightarrow+\infty}\frac{{\max_{t \in [-\xi, \xi]}H(t)}}{\xi^2}=1,\,\,\,\,\,\,\,\limsup_{\xi\rightarrow +\infty}\frac{H(\xi)}{\xi^2}=+\infty.
$$
\noindent Taking into account Remark \ref{general}, for each
parameter $\lambda$ belonging to
$$
\left]0,\frac{1}{2(2N+3)^2}\right[,
$$
\noindent the following problem
\begin{equation}\tag{$S_{\lambda}^{h}$}\label{N3}
\left\{
\begin{array}{l}
\Delta u(x)+\lambda h(u(x))=u(x)\quad x \in V\setminus V_0, \\
u|_{V_0}=0,\\
\end{array}
\right.
\end{equation}
\noindent possesses an unbounded sequence of strong solutions.}
\end{example}



\medskip
 {\bf Acknowledgements.}  V.~R\u adulescu acknowledges the support through  Grant CNCSIS PNII--ID--PCE--47/2011.

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