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\title[The Dirichlet problem on the Sierpinski gasket]{Infinitely many solutions for the Dirichlet problem on the Sierpinski gasket}

\author[B. E. Breckner]{Brigitte E. Breckner}
\address{Babe\c{s}-Bolyai University, Faculty of Mathematics and Computer Science\\
Kogalniceanu str. 1\\
400084 Cluj-Napoca, Romania} \email{breckner@gmx.net}

\author[V. R\u adulescu]{Vicen\c tiu R\u adulescu}
\address{Institute of Mathematics ``Simion Stoilow" of the Romanian
Academy, 010702 Bucharest, Romania} \email{vicentiu.radulescu@imar.ro}

\author[Cs. Varga]{Csaba Varga}
\address{Babe\c{s}-Bolyai University, Faculty of Mathematics and Computer Science\\
Kogalniceanu str. 1\\
400084 Cluj-Napoca, Romania} \email{varga\_gy\_csaba@yahoo.com}

\date{}


\begin{abstract}
Adapting a method from \cite{FarKr}, used therein in the context of
Sobolev spaces, to the case of certain spaces of functions on
fractal domains, we prove that the following nonlinear elliptic
equation, defined on the Sierpinski gasket,
$$\Delta u(x)+a(x)u(x)=g(x)f(u(x))$$
and with zero Dirichlet boundary conditions has infinitely many weak
solutions.
\end{abstract}

\subjclass[2000]{Primary 35J20; Secondary 28A80, 35J25, 35J60,
47J30, 49J52}

\keywords{Sierpinski gasket, weak Laplacian, nonlinear elliptic
equation on the Sierpinski gasket, weak solution}


\maketitle

\section{Introduction}
Many physical problems on fractal domains lead to nonlinear models
(for example, reaction-diffusion equations, problems on elastic
fractal media or fluid flow through fractal regions), so it is
appropriate to study nonlinear partial differential equations on
fractals. In the last years there has been an increasing interest in
studying such equations, also motivated and stimulated by the
considerable amount of literature devoted to the definition of a
Laplacian operator for functions on fractal domains. Among the
contributions to the theory of nonlinear elliptic equations on
fractals we mention \cite{Fa99}, \cite{FaHu}, \cite{Hu},
\cite{HuaZh}, \cite{stripaper}. 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 minimax results (mountain pass
theorems, saddle-point theorems), respectively, mi\-ni\-mi\-zation
procedures. In the present paper, we propose another method to study
this type of equations. To apply this method we have to impose on
the nonlinear term of the elliptic equation other conditions as
those used in the papers mentioned before. Instead of requiring that
the nonlinear term should satisfy certain symmetry properties, this
term has to have an oscillating behavior. This method has been used
successfully to prove, in the context of certain Sobolev spaces, the
existence of infinitely many solutions for Dirichlet problems on
bounded domains \cite{Sa}, for one-dimensional scalar field
equations and systems \cite{FarKr}, for homogeneous Neumann problems
\cite{KrMo}. The aim of the present paper is to show that the methods used
in \cite{FarKr} can be successfully adapted to prove the existence
of infinitely many (weak) solutions for nonlinear elliptic equations
with zero Dirichlet boundary conditions on the Sierpinski gasket.
\medskip

\noindent {\bf Notations.} We denote by $\N$ the set of natural
numbers $\{0,\, 1,\, 2,\dots\}$, by $\N^*:=\N\setminus\{0\}$ the set
of positive naturals, and by $|\cdot|$ the Euclidian norm on the
spaces $\R^n$, $n\in\N^*$.


\section{The Sierpinski gasket}
In its initial representation that goes back to the pioneering papers of the Polish mathematician Waclaw Sierpinski (1882--1969), the {\it Sierpinski gasket} is the connected subset
of the plane obtained from an equilateral
triangle by removing the open middle inscribed
equilateral triangle of $4^{-1}$ the area, removing
the corresponding open triangle from each of
the three constituent triangles, and continuing
this way. The gasket can also be obtained as
the closure of the set of vertices arising in this
construction. Over the years, the Sierpinski gasket showed both to be
extraordinarily useful in representing roughness in nature
and man's works. 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. We also refer to the excellent monograph Ciarlet \cite{pgcbook} for a thorough introduction to relevant applications of the nonlinear analysis.

We now rigorously describe the construction of the Sierpinski gasket in a general setting.
Let $N\geq2$ be a natural number and let $p_1,\dots, p_N\in\R^{N-1}$
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
$$S_i(x)=\frac12\,x+\frac12\,p_i\,.$$
Obviously every $S_i$ is a similarity with ratio $\frac12$. Let
${\mathcal S}:=\{S_1,\dots, S_N\}$ and denote by $F\colon{\mathcal
P}(\R^{N-1})\to{\mathcal P}(\R^{N-1})$ the map assigning to a subset
$A$ of $\R^{N-1}$ the set
$$F(A)=\bigcup_{i=1}^NS_i(A).$$
It is known (see, for example, Theorem 9.1 in \cite{Fa}) that there
is a unique non-empty compact subset $V$ of $\R^{N-1}$, called the
{\it attractor of the family} ${\mathcal S}$, such that $F(V)=V$
(that is, $V$ is a fixed point of the map $F$). The set $V$ is called
the {\it Sierpinski gasket} (SG for short) in $\R^{N-1}$. It can be
constructed inductively as follows: Put $V_0:=\{p_1,\dots,p_N\}$,
$V_m:=F(V_{m-1})$, for $m\geq1$, and $V_*:=\cup_{m\geq0}V_m$. Since
$p_i=S_i(p_i)$ for $i=\overline{1,N}$, we have $V_0\subseteq V_1$,
hence $F(V_*)=V_*$. Taking into account that the maps $S_i$,
$i=\overline{1,N}$, are homeomorphisms, we conclude that
$\overline{V_*}$ is a fixed point of $F$. 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 $i=\overline{1,N}$. 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
$V=\overline{V_*}$. 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 {\it intrinsic boundary} of
the SG.

The family ${\mathcal S}$ of similarities satisfies the open set
condition (see pg. 129 in \cite{Fa}) with the interior $\inter\, C$
of $C$. (Note that $\inter\, C\neq\emptyset$ since the points
$p_1,\dots, p_N$ are affine independent.) Thus, by Theorem 9.3 of
\cite{Fa}, the Hausdorff dimension $d$ of $V$ satisfies the equality
$$\sum_{i=1}^N\left(\frac12\right)^d=1,$$
hence $d=\frac{\ln N}{\ln 2}$, and $0<{\mathcal H}^d(V)<\infty$,
where ${\mathcal H}^d$ is the $d$-dimensional Hausdorff measure on
$\R^{N-1}$. Let $\mu$ be the normalized restriction of ${\mathcal
H}^d$ to the subsets of $V$, so $\mu(V)=1$. The following property
of $\mu$ will be important for the proof of the main result
\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 $V$. To prove
(\ref{support}), let $B$ be a non-empty open subset of $V$ and fix
an arbitrary element $x\in B$. Then (see 3.1 (iii) in \cite{Hut})
the equality $F(V)=V$ yields the existence of a function
$\phi\colon\N^*\to\{1,\dots,N\}$ such that $x$ is the unique element
in the intersection of the members of the following sequence of sets
$$V\supseteq V_{i_1}\supseteq V_{i_1i_2}\supseteq \dots \supseteq V_{i_1i_2\dots i_n}\supseteq\dots,$$
where $V_{i_1\dots i_n}:=(S_{\phi(1)}\circ\dots\circ
S_{\phi(n)})(V)$ for every $n\in\N^*$. Assuming that
$$V_{i_1\dots i_n}\setminus B\neq\emptyset, \hbox{ for every }n\in\N^*,$$
 there exists an element $x_n\in V_{i_1\dots i_n}\setminus B$ for every $n\in\N^*$. Since
$$|x_n-x|\leq\diam\, V_{i_1\dots i_n}=\left(\frac12\right)^n\diam\, V,\hbox{ for all }n\in\N^*,$$
the sequence $(x_n)$ converges to $x$. Thus there is an index $n_0$
with $x_n\in B$ for all $n\geq n_0$, a contradiction. We conclude
that there is  $n\in\N^*$ such that
$$V_{i_1\dots i_n}\subseteq B.$$
It follows that $\mu(V_{i_1\dots i_n})\leq\mu(B)$. On the other
hand, by the scaling property of the Hausdorff measure (see 2.1 in
\cite{Fa}), we have that
$$\mu(V_{i_1\dots i_n})=\left(\frac12\right)^{nd}\cdot\mu(V)>0,$$
so $\mu(B)>0$.



\section{The space $H_0^1(V)$} We retain the notations from the previous section and briefly recall from \cite{FaHu} the following notions
(see also \cite{Hu} and \cite{Ko} for the case $N=3$). 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 $||\cdot||_s$. 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 very 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 usual $||\cdot||_2$ 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
$${\mathcal W}(u,v)=\lim_{m\to\infty} {\mathcal W}_m(u,v).$$
Then ${\mathcal W}(u,v)\in\R$ and $H_0^1(V)$, equipped with the
inner product ${\mathcal W}$ (which obviously induces the norm
$||\cdot||$), becomes a real Hilbert space. Moreover,
\begin{equation}\label{embeddingconstant}
||u||_s\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||_s)
\end{equation}
is compact.

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 that $H_0^1(V)$ is dense in $L^2(V,\mu)$, that ${\mathcal
W}$ is a Dirichlet form on $L^2(V,\mu)$. (In fact, it is the
analogue in the case of $H_0^1(V)$ of a property stated in
\cite{MaMi} for Sobolev spaces.)

\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 $||h\circ u||\leq L\cdot||u||$.
\end{lemma}
\begin{proof}
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
$$W_m(h\circ u)\leq L^2\cdot W_m(u).$$
Hence $W(h\circ u)\leq L^2\cdot W(u)$, according to (\ref{defW}).
Thus $h\circ u\in H_0^1(V)$ and $||h\circ u||\leq L\cdot||u||$.
\end{proof}

\section{The Dirichlet problem on the Sierpinski gasket} Keep the notations from the previous sections.
We also recall from \cite{FaHu} (respectively, from \cite{Hu} and \cite{Ko} in the case $N=3$) that
one can define in a standard way a linear self-adjoint operator $\Delta\colon D\to L^2(V,\mu)$,
where $D$ 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
$$-{\mathcal W}(u,v)=\int_V\Delta u\cdot vd\mu,\hbox{ for every }(u,v)\in D\times H_0^1(V).$$
The operator $\Delta$ is called the {\it weak Laplacian on} $V$.

Given $a\colon V\to\R$, $f\colon\R\to\R$ and $g\colon V\to\R$, we
can formulate now the following {\it Dirichlet problem on the SG}:
Find  functions $u\in H_0^1(V)$ such that
$$(P)
\left\{
\begin{array}{l}
\Delta u(x)+a(x)u(x)=g(x)f(u(x)),\ \forall x\in V\setminus V_0,\\
\\
u|_{V_0}=0.
\end{array}
\right.
$$
A function $u\in H_0^1(V)$ is called a {\it weak solution of} $(P)$
if
$${\mathcal W}(u,v)-\int_Va(x)u(x)v(x)d\mu+\int_Vg(x)f(u(x))v(x)d\mu=0,\ \forall v\in H_0^1(V).$$
The aim of the paper is to prove the following result concerning the
existence of multiple weak solutions of the problem $(P)$:

\begin{theorem}\label{main}
Assume that the following conditions hold:
\begin{itemize}
\item[{\rm (C1)}] $a\in L^1(V,\mu)$ and $a\leq0$ a.e. in $V$.
\item[{\rm (C2)}] $f\colon\R\to\R$ is continuous such that
\begin{itemize}
\item[{\rm (1)}] There exist two sequences $(a_k)$ and $(b_k)$ in $]0,\infty[$ with $b_{k+1}<a_k<b_{k}$,
$\displaystyle\lim_{k\to \infty}b_k=0$ and such that $f(s)\leq 0
\hbox{ for every }s\in [a_k,b_k].$
\item[{\rm (2)}] Either $\sup\{s<0\mid f(s)>0\}=0$, or there is a
$\delta>0$ with $f|_{[-\delta,0]}= 0$.
\end{itemize}
\item[{\rm (C3)}] $F\colon\R\to\R$, defined by $F(s)=\int_0^s f(t)dt$, is such that
\begin{itemize}
\item[{\rm (3)}] $\displaystyle-\infty<\liminf_{s\to 0^+}\frac{F(s)}{s^2}$,
\item[{\rm (4)}] $\displaystyle\limsup_{s\to 0^+}\frac{F(s)}{s^2}=\infty$.
\end{itemize}
\item[{\rm (C4)}] $g\in C(V)$ with $g\leq0$ and such that the restriction of $g$ to every open subset of $V$ is not identically 0.
\end{itemize}
Then there is a sequence $(u_k)$ of pairwise distinct weak solutions
of problem $(P)$ such that
$\displaystyle\lim_{k\to\infty}\|u_k\|=0.$ In particular,
$\displaystyle\lim_{k\to\infty}\|u_k\|_s=0.$
\end{theorem}

\begin{remark}
The conditions (1) and (2) of Theorem \ref{main} on the nonlinear
term $f\colon\R\to\R$ of problem $(P)$ show that this function has
an oscillating behavior at $0^+$. In \cite{FarKr} there is given the
following example for a function satisfying the conditions $(C2)$
and $(C3)$ of the theorem: Let $0<\alpha<1<\beta$ and
$f\colon\R\to\R$ be such that $f(0)=0$ and
$f(t)=|t|^\alpha\max\{0,\sin|t|^{-1}\}+|t|^\beta\min\{0,\sin|t|^{-1}\}$
for $t\neq 0$.
\end{remark}


\section{Preparatory results} Let $a, g\in L^1(V,\mu)$ and $f\colon\R\to\R$ be continuous. Define $F\colon\R\to\R$ by $F(s)=\int_0^s f(t)dt$.
The map $I\colon H_0^1(V)\to\R$  defined by
\begin{equation}\label{energyfunctional}
I(u)=\frac12||u||^2-\frac12\int_Va(x)u^2(x)d\mu+\int_Vg(x)F(u(x))d\mu,\
\forall u\in H_0^1(V),
\end{equation}
will turn out to be under suitable assumptions the energy functional
attached to problem $(P)$ (see Proposition \ref{criticalpoint}
below). To see this we first recall a few basic notions.

\begin{definition}
Let $E$ be a real Banach space 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 $dT(u)\colon E\to\R$, called
the {\it Fr\'{e}chet differential of $T$ at $u$},  such that
$$\lim_{v\to 0}\frac{|T(u+v)-T(u)-dT(u)(v)|}{||v||}=0.$$
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 $dT(u)=0$.
\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}

\begin{proposition}\label{criticalpoint}
Suppose that $a\in L^1(V,\mu)$, $f\colon\R\to\R$ is continuous and
$g\in C(V)$. Then the functional $I\colon H_0^1(V)\to\R$ defined by
{\rm (\ref{energyfunctional})} is Fr\'{e}chet differentiable on
$H_0^1(V)$. Moreover, for every $u,\,v\in H_0^1(V)$ the following
equality holds
$$dI(u)(v)={\mathcal W}(u,v)-\int_Va(x)u(x)v(x)d\mu+\int_Vg(x)f(u(x))v(x)d\mu.$$
In particular, $u\in H_0^1(V)$ is a weak solution of problem $(P)$
if and only if $u$ is a critical point of $I$.
\end{proposition}
\begin{proof} See Proposition 2.19 in \cite{FaHu}.
\end{proof}


\begin{remark}\label{strong}
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 \cite{FaHu}, it
follows that every weak solution of the problem $(P)$ is also a
strong solution (as defined in \cite{FaHu}).
\end{remark}

We next do some preparation in order to state in Proposition
\ref{minimum} an important property of the map $I$.

\begin{definition}
Let $X$ be a topological Hausdorff space. A map $h\colon X\to\R$ is
called {\it sequentially lower semicontinuous at $x\in X$} if for
every sequence $(x_n)$ in $X$ converging to $x$ the inequality
$$h(x)\leq\liminf_{n\to\infty} h(x_n)$$
holds. The map $h\colon X\to\R$ is {\it sequentially lower
semicontinuous on $X$} if $h$ is sequentially lower semicontinuous
at every point $x\in X$.
\end{definition}

\begin{lemma}\label{Iwslsc}
Let $a,\, g\in L^1(V,\mu)$ and let $f\colon\R\to\R$ be continuous.
Then the functional $I\colon H_0^1(V)\to\R$ defined by {\rm
(\ref{energyfunctional})} is weakly sequentially lower
semicontinuous on $H_0^1(V)$.
\end{lemma}
\begin{proof}
We first prove that the maps $u\in
H_0^1(V)\mapsto\int_Va(x)u^2(x)d\mu\in\R$ and $u\in
H_0^1(V)\mapsto\int_Vg(x)F(u(x))d\mu\in\R$ are both weakly
sequentially continuous on $H_0^1(V)$. To this end let $u\in
H_0^1(V)$ and consider a sequence $(u_n)$ weakly converging to $u$
in $H_0^1(V)$. Since the embedding (\ref{embedding}) is compact, the
sequence $(u_n)$ converges to $u$ in $C_0(V)$, and so does $(u_n^2)$
to $u^2$. The relations
\begin{eqnarray*}
\left|\int_Va(x)u^2_n(x)d\mu-\int_Va(x)u^2(x)d\mu\right|&\leq&\int_V|a(x)|\cdot|u^2_n(x)-u^2(x)|d\mu\\
&\leq&||u_n^2-u^2||_s\cdot\int_V|a(x)|d\mu
\end{eqnarray*}
show that the map $u\in H_0^1(V)\mapsto\int_Va(x)u^2(x)d\mu\in\R$ is
weakly sequentially continuous at $u$, hence on $H_0^1(V)$, because
$u$ was arbitrarily chosen.

Since $(u_n)$ converges to $u$ in $C_0(V)$ there is a positive
constant $\rho$ such that $||u||_s\leq \rho$ and $||u_n||_s\leq
\rho$ for all $n\in\N$. Denote by
$c:=\displaystyle\max_{t\in[-\rho,\rho]}|f(t)|$. Then, for every
$n\in\N$,
$$|F(u_n(x))-F(u(x))|=\left|\int_{u(x)}^{u_n(x)}f(t)dt\right|\leq c\cdot||u_n-u||_s,$$
hence
$$\left|\int_Vg(x)F(u_n(x))d\mu-\int_Vg(x)F(u(x))d\mu\right|\leq c\cdot||u_n-u||_s\cdot\int_V|g(x)|d\mu.$$
We conclude as above that the map $u\in
H_0^1(V)\mapsto\int_Vg(x)F(u(x))d\mu\in\R$ is weakly sequentially
continuous on $H_0^1(V)$.

The map $u\in H_0^1(V)\mapsto||u||^2\in\R$ is continuous in the norm
topology on $H_0^1(V)$ and convex, thus it is weakly sequentially
lower semicontinuous on $H_0^1(V)$. We conclude that $I$ is also
weakly sequentially lower semicontinuous on $H_0^1(V)$.
\end{proof}

\begin{lemma}\label{auxiliary}
Let $X$ be a non-empty topological Hausdorff space and let $h\colon
X\to\R$ be a map satisfying the following properties
\begin{itemize}
\item[{\rm(i)}] $h$ is sequentially lower semicontinuous on $X$,
\item[{\rm(ii)}] $h$ is bounded from below on $X$,
\item[{\rm(iii)}] every sequence $(x_n)$ in $X$ such that the sequence $(h(x_n))$ is bounded has a convergent subsequence.
\end{itemize}
Then $h$ attains its infimum on $X$.
\end{lemma}
\begin{proof}
Let $(x_n)$ be a sequence in $X$ such that $\lim h(x_n)=\inf h(X)$.
According to (ii), the sequence $(h(x_n))$ is bounded, so, by (iii),
there is a convergent subsequence $(x_{n_k})$ of $(x_n)$. Let $x\in
X$ be the limit point of this subsequence. Using (i), we get
$$h(x)\leq \liminf_{k\to\infty}h(x_{n_k})=\inf h(X),$$
hence $h(x)=\inf h(X)$.
\end{proof}

\begin{remark}\label{inequalities}
Let $a,\, g\in L^1(V,\mu)$ be so that $a\leq0$ and $g\leq0$ a.e. in
$V$, and let $f\colon\R\to\R$ be continuous. Consider $u\in
H_0^1(V)$ and $d,b\in\R$ such that $d\leq u(x)\leq b$ for every
$x\in V$.  According to the fact that $g\leq0$ a.e. in $V$, we then
have
\begin{equation}\label{inequ1}
\int_Vg(x)F(u(x))d\mu\geq \max_{s\in[d,b]}F(s)\cdot\int_Vg(x)d\mu.
\end{equation}
For later use we state the following relations concerning the
functional $I\colon H_0^1(V)\to\R$ defined by {\rm
(\ref{energyfunctional})}: The inequalities (\ref{inequ1}) and
$a\leq0$ a.e. in $V$ imply that
\begin{equation}\label{inequ2}
I(u)\geq \max_{s\in[d,b]}F(s)\cdot\int_Vg(x)d\mu
\end{equation}
and
\begin{equation}\label{inequ3}
\frac12||u||^2\leq I(u)- \max_{s\in[d,b]}F(s)\cdot\int_Vg(x)d\mu.
\end{equation}
We also note that for every $x\in V$
$$F(u(x))\leq |F(u(x))|=\left|\int_0^{u(x)}f(t)dt\right|\leq \max_{t\in[d,b]}|f(t)|\cdot||u||_s.$$
As above we then conclude that
\begin{equation}\label{inequ4}
I(u)\geq \max_{t\in[d,b]}|f(t)|\cdot||u||_s\cdot \int_Vg(x)d\mu
\end{equation}
and
\begin{equation}\label{inequ5}
\frac12||u||^2\leq I(u)-\max_{t\in[d,b]}|f(t)|\cdot||u||_s\cdot
\int_Vg(x)d\mu.
\end{equation}
\end{remark}

\begin{proposition}\label{minimum}
Let $a,\, g\in L^1(V,\mu)$ be so that $a\leq0$ and $g\leq0$ a.e. in
$V$, and let $f\colon\R\to\R$ be continuous. Consider $d,b\in\R$ so
that $d<0<b$ and put
$$M:=\{u\in H_0^1(V)\mid d\leq u(x)\leq b,\ \forall x\in V\}.$$
Then the functional $I\colon H_0^1(V)\to\R$ defined by {\rm
(\ref{energyfunctional})} is bounded from below on $M$ and attains
its infimum on $M$.
\end{proposition}
\begin{proof}
Obviously the set $M$ is non-empty (it contains the constant 0
function) and convex. Since the inclusion (\ref{embedding}) is
continuous, $M$ is closed in the norm topology on $H_0^1(V)$. It
follows that $M$ is also closed in the weak topology on $H_0^1(V)$.
Consider $M$ be endowed with the relative weak topology. We will
show that the restriction $I|_M$ of $I$ to $M$ satisfies the
hypotheses (i)--(iii) of Lemma \ref{auxiliary}. Condition (i)
results from Lemma \ref{Iwslsc}, and (ii) is a consequence of
(\ref{inequ2}). Hence $I$ is bounded from below on $M$. To verify
(iii), let $(u_n)$ be a sequence in $M$ such that the sequence
$(I(u_n))$ is bounded. Inequality (\ref{inequ3}) then yields  that
$(u_n)$ is (norm) bounded. By the reflexivity of the Hilbert space
$H_0^1(V)$ and the closedness of $M$ in the weak topology we
conclude that the sequence $(u_n)$ has a weakly convergent
subsequence in $M$. Lemma \ref{auxiliary} implies now that $I$
attains its infimum on $M$.
\end{proof}


\section{Proof of Theorem \ref{main}} Throughout this section we assume that the conditions $C(1)-C(4)$ in the hypotheses of Theorem \ref{main}
are satisfied.

{\it Case 1}: Suppose first that the equality $\sup\{s<0\mid
f(s)>0\}=0$ in condition (2) of $(C2)$ holds. Then there exists a
strictly increasing sequence $(c_k)$ of negative reals such that
$\lim c_k=0$ and $f(c_k)>0$ for every natural $k$. By continuity of
$f$ there exists another sequence $(d_k)$ such that
$d_k<c_k<d_{k+1}$ and $f(t)>0$ for every $t\in[d_k,c_k]$ and every
natural $k$.

{\it Case 2}: If we have in (2) that there is a $\delta>0$ with
$f|_{[-\delta,0]}= 0$, then choose a strictly increasing sequence
$(c_k)$ of negative reals greater than $-\delta$ such that $\lim
c_k=0$. Construct the sequence $(d_k)$ such that $d_k<c_k<d_{k+1}$
and $f(t)=0$ for every $t\in[d_k,c_k]$ and every natural $k$.

In both cases, since $F(s)=\int_0^sf(t)dt$ for every $s\in\R$, it
follows that
\begin{equation}\label{Fdc}
F(s)\leq F(c_k),\hbox{ for every }s\in[d_k,c_k].
\end{equation}
Using condition (1) of $(C2)$ we have that
\begin{equation}\label{Fab}
F(s)\leq F(a_k),\hbox{ for every }s\in[a_k,b_k].
\end{equation}

For every $k\in\N$ set now
$$M_k:=\{u\in H_0^1(V)\mid d_k\leq u(x)\leq b_k,\ \forall x\in V\}.$$
The proof of Theorem \ref{main} includes the following main steps
contained in the next lemmata:

(i) we  show that the map $I\colon H_0^1(V)\to\R$ defined
by {\rm (\ref{energyfunctional})} has at least one critical point in
each of the sets $M_k$;

(ii) we will show that there are infinitely many pairwise
distinct such critical points,

(iii) by Proposition \ref{criticalpoint} we know that each of
these critical points is a weak solution of Problem $(P)$.

\begin{lemma}\label{minimumk}
For every $k\in\N$ there is an element $u_k\in M_k$ such that the
following conditions hold:
\begin{itemize}
\item[{\rm (i)}] $I(u_k)=\inf I(M_k)$,
\item[{\rm (ii)}] $c_k\leq u_k(x)\leq a_k$, for every $x\in V$.
\end{itemize}
\end{lemma}
\begin{proof}
Fix $k\in\N$. According to Proposition \ref{minimum}, there is an
element $\tilde u_k\in M_k$ such that $I(\tilde u_k)=\inf I(M_k)$.
Define $h\colon\R\to\R$ by
$$h(t)\left\{
\begin{array}{l}
c_k,\ t<c_k\\
x,\ t\in[c_k,a_k]\\
a_k,\ t>a_k.
\end{array}
\right.
$$
Note that $h(0)=0$ and that $h$ is a Lipschitz map with Lipschitz
constant $L=1$. According to Lemma \ref{MaMianalogue} the map
$u_k:=h\circ \tilde u_k$ belongs to $H_0^1(V)$ and
\begin{equation}\label{mic1}
||u_k||\leq ||\tilde u_k||.
\end{equation}
Moreover, $u_k$ belongs to $M_k$ and obviously satisfies condition
(ii) to be proved. We next show that (i) also holds. For this set
$$V_1:=\{x\in V\mid \tilde u_k(x)<c_k\},\quad V_2:=\{x\in V\mid \tilde u_k(x)>a_k\}.$$
Then
$$
u_k(x)= \left\{
\begin{array}{l}
c_k,\ x\in V_1\\
\tilde u_k(x),\ x\in V\setminus(V_1\cup V_2)\\
a_k,\ x\in V_2.
\end{array}
\right.
$$
It follows that
\begin{equation}\label{mic2}
(u_k(x))^2\leq (\tilde u_k(x))^2,\hbox{ for every }x\in V.
\end{equation}
Furthermore, if $x\in V_1$ then $\tilde u_k(x)\in[d_k, c_k[$, hence
$F(\tilde u_k(x))\leq F(c_k)=F(u_k(x))$, by (\ref{Fdc}).
Analogously, if $x\in V_2$, then (\ref{Fab}) yields $F(\tilde
u_k(x))\leq F(a_k)=F(u_k(x))$. Thus
\begin{equation}\label{mic3}
F(\tilde u_k(x))\leq F(u_k(x)),\hbox{ for every }x\in V.
\end{equation}
The inequalities (\ref{mic1}), (\ref{mic2}) and (\ref{mic3}) imply,
together with the fact that $a\leq0$ and $g\leq0$ a.e. in $V$, that
\begin{eqnarray*}
I(\tilde u_k)-I(u_k)&=&\frac12 ||\tilde u_k||^2-\frac12 ||u_k||^2-\frac12\int_Va(x)(\tilde u_k^2(x)-u_k^2(x))d\mu\\
&+&\int_Vg(x)(F(\tilde u_k(x))-F(u_k(x)))d\mu \geq0.
\end{eqnarray*}
Thus $I(\tilde u_k)\geq I(u_k)$. Since $I(\tilde u_k)=\inf I(M_k)$
and since $u_k\in M_k$, we conclude that $I(u_k)=\inf I(M_k)$, thus
(i) is also fulfilled.
\end{proof}

\begin{lemma}\label{criticalk}
For every $k\in\N$ let $u_k\in M_k$ be a function satisfying the
conditions {\rm (i)} and {\rm (ii)} of Lemma {\rm \ref{minimumk}}.
The functional $I$ has then in $u_k$ a local minimum (with respect
to the norm topology on $H_0^1(V)$), for every $k\in\N$. In
particular, $(u_k)$ is a sequence of weak solutions of problem
$(P)$.
\end{lemma}
\begin{proof}
Fix $k\in\N$. Suppose to the contrary that $I$ has not in $u_k$ a
local minimum. This implies the existence of a sequence $(w_n)$ in
$H_0^1(V)$ converging to $u_k$ in the norm topology such that
$$ I(w_n)<I(u_k),\hbox{ for every }n\in\N.$$
In particular, $w_n\notin M_k$, for all $n\in\N$. Choose a real
number $\varepsilon$ such that
$0<\varepsilon<\frac12\min\{b_k-a_k,c_k-d_k\}$. In view of
(\ref{embeddingconstant}) the sequence $(w_n)$ converges to $u_k$ in
the supremum norm topology on $C(V)$. Hence there is an index
$m\in\N$ such that
$$||w_m-u_k||_s\leq\varepsilon.$$
For every $x\in V$ we then have according to (ii) of Lemma
\ref{minimumk}
$$w_m(x)=w_m(x)-u_k(x)+u_k(x)\leq\varepsilon+u_k(x)\leq \frac{b_k-a_k}{2}+a_k<b_k$$
and
$$w_m(x)=w_m(x)-u_k(x)+u_k(x)\geq-\varepsilon+u_k(x)\geq \frac{d_k-c_k}{2}+c_k>d_k.$$
Thus $w_m\in M_k$, a contradiction. We conclude that $I$ has in
$u_k$ a local minimum. The last assertion of the lemma follows now
from Remark \ref{useful} and Proposition \ref{criticalpoint}.
\end{proof}

\begin{lemma}\label{infinite}
For every $k\in\N$ put $\gamma_k:=\inf I(M_k)$. Then $\gamma_k<0$
for all $k\in\N$ and $\displaystyle\lim_{k\to\infty}\gamma_k=0$.
\end{lemma}
\begin{proof}
Fix a function $u\in H_0^1(V)$ 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 subset of $V$. Let $h\colon\R\to\R$ be defined by
$h(t)=|\min\{t,1\}|$, for every $t\in\R$. Then $h(0)=0$ and $h$ is a
Lipschitz map with Lipschitz constant $L=1$. Lemma
\ref{MaMianalogue} implies that $v:=h\circ u\in H_0^1(V)$. Moreover,
$v(x)=1$ for every $x\in D$, and $0\leq v(x)\leq1$ for every $x\in
V$.

On the other hand, condition (3) of $(C3)$ implies the existence of
real numbers $\rho>0$ and $c$ such that $\frac{F(s)}{s^2}>c$ for
every $s\in]0,\rho[$. It follows that
\begin{equation}\label{mic4}
F(s)\geq cs^2,\hbox{ for every } s\in[0,\rho[.
\end{equation}
Condition (4) of $(C3)$ yields the existence of a sequence $(r_n)$
in $]0,\rho[$ such that $\displaystyle\lim_{n\to\infty}r_n=0$ and
\begin{equation}\label{mic5}
\lim_{n\to\infty}\frac{F(r_n)}{r_n^2}=\infty.
\end{equation}
We then have for every $n\in\N$
\begin{eqnarray*}
I(r_nv)=\frac{r_n^2}{2}||v||^2-\frac{r_n^2}{2}\int_Va(x)v^2(x)d\mu&+&F(r_n)\int_Dg(x)d\mu\\
&+&\int_{V\setminus D}g(x)F(r_nv(x))d\mu.
\end{eqnarray*}
Using (\ref{mic4}) and the fact that $g\leq0$ in $V$, we get for
every $n\in\N$
\begin{eqnarray*}
I(r_nv)\leq \frac{r_n^2}{2}||v||^2-\frac{r_n^2}{2}\int_Va(x)v^2(x)d\mu&+&F(r_n)\int_Dg(x)d\mu\\
&+&cr_n^2\int_{V\setminus D}g(x)v^2(x)d\mu,
\end{eqnarray*}
thus
$$\frac{I(r_nv)}{r_n^2}\leq \frac12||v||^2-\frac12\int_Va(x)v^2(x)d\mu+\frac{F(r_n)}{r_n^2}\int_Dg(x)d\mu+c\int_{V\setminus D}g(x)v^2(x)d\mu.$$
Condition $(C4)$ and (\ref{support}) imply that $\int_Dg(x)d\mu<0$,
so we get from (\ref{mic5}) and the above inequality that
$$\lim_{n\to\infty}\frac{I(r_nv)}{r_n^2}=-\infty.$$
Thus there is an index $n_0$ such that $I(r_nv)<0$ for every $n\geq
n_0$. Fix now $k\in\N$. Since
$\displaystyle\lim_{n\to\infty}||r_nv||_s=0$, we get an index $p\geq
n_0$ such that $r_pv\in M_k$. Hence $\gamma_k\leq I(r_pv)<0$.

Let $u_k\in M_k$ be so that $\gamma_k=I(u_k)$. Since $M_k\subseteq
M_0$, relation (\ref{inequ4}) yields
$$\gamma_k=I(u_k)\geq \max_{t\in[d_0,b_0]}|f(t)|\cdot||u_k||_s\cdot \int_Vg(x)d\mu,$$
hence
$$0>\gamma_k\geq \max_{t\in[d_0,b_0]}|f(t)|\cdot\max\{b_k,|d_k|\} \cdot \int_Vg(x)d\mu.$$
Since $\displaystyle\lim_{k\to\infty} b_k=\lim_{k\to\infty} d_k=0$,
we conclude that $\displaystyle\lim_{k\to\infty}\gamma_k=0$.
\end{proof}

\medskip
{\it Proof of Theorem \ref{main} concluded}. From Lemma
\ref {criticalk} we know that there is a sequence $(u_k)$ of weak
solutions of problem $(P)$ such that $\gamma_k=I(u_k)$, where
$\gamma_k=\inf I(M_k)$, for every natural $k$. Using relation
(\ref{inequ5}) and the fact that $\gamma_k\leq0$, we obtain
$$\frac{1}{2}||u_k||^2\leq -\max_{t\in[d_0,b_0]}|f(t)|\cdot\max\{b_k,|d_k|\} \cdot \int_Vg(x)d\mu.$$
Using once again that $\displaystyle\lim_{k\to\infty}
b_k=\lim_{k\to\infty} d_k=0$, we conclude that
$\displaystyle\lim_{k\to\infty} ||u_k||=0$. Thus also
$\displaystyle\lim_{k\to\infty} ||u_k||_s=0$, by
(\ref{embeddingconstant}).

We know from Lemma \ref{infinite} that  $I(u_k)=\gamma_k<0$, for
every natural $k$, and that $\displaystyle\lim_{k\to\infty}
\gamma_k=0$. Thus we can find a subsequence $(u_{k_j})$ of the
sequence $(u_k)$ consisting of pairwise distinct elements. $\square$

\begin{remark}
1) If to the hypotheses of Theorem \ref{main} one adds the
requirement that $a\in C(V)$, then Theorem \ref{main} and Remark
\ref{strong} yield the existence of a sequence of pairwise distinct
strong solutions of problem $(P)$ converging to $0$.

2) By the same method one can prove (see also \cite{FarKr}) an
analogous result in the case the nonlinear term $f\colon\R\to\R$ has
an oscillating behavior at $\infty$. In this case one obtains a
sequence $(u_k)$ of pairwise distinct weak solutions of problem
$(P)$ such that $\displaystyle\lim_{k\to\infty}||u_k||=\infty$.
\end{remark}




\subsection*{Acknowledgements}
B. Breckner and Cs. Varga have been supported by the grants PN-II-ID-527
and PN-II-P4-11-020/2007.

The authors also thank Professor Jiaxin Hu for his helpful
suggestions.



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\end{document}
