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\begin{document}

\begin{frontmatter}


\selectlanguage{english}
\title{Infinitely many solutions for a class of nonlinear elliptic problems on fractals}

\vspace{-2.6cm}

\selectlanguage{french}
\title{Infinit\'e de solutions pour une classe de probl\`emes elliptiques non lin\'eaires sur des fractales}

\selectlanguage{english}
\author[GB]{Gabriele Bonanno}
\ead{bonanno@unime.it}
\author[GM]{Giovanni Molica Bisci}
\ead{gmolica@unirc.it}
\author[VR1,VR2]{Vicen\c tiu R\u{a}dulescu}
\thanks[label3]{Correspondence address: Vicen\c tiu R\u{a}dulescu, University of Craiova,
Department of Mathematics,
200585 Craiova, Romania. E-mail: {\tt
vicentiu.radulescu@imar.ro}}
\ead{vicentiu.radulescu@imar.ro}
\ead[url]{http://www.inf.ucv.ro/\~{}radulescu}
\address[GB]{Department of Science for Engineering and Architecture (Mathematics Section),
Engineering Faculty,\\ University of Messina,
98166 Messina, Italy}
\address[GM]{Dipartimento MECMAT, University of Reggio Calabria,
 Via Graziella, Feo di Vito, 89124 Reggio Calabria, Italy}
\address[VR1]{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,  014700
Bucharest, Romania}
\address[VR2]{Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585 Craiova, Romania}

\begin{abstract}
% Text of abstract in English
We study the nonlinear problem
$\Delta u+a(x)u=\lambda g(x)f(u)$ in $V\setminus V_0$,
$u=0$ on $V_0$, where $V$ is the Sierpi\'nski gasket, $V_0$ is its intrinsic boundary, $\Delta$ denotes the weak Laplace operator, $\lambda$ is a positive parameter, and $f$ has an oscillatory behaviour either near the origin or at infinity. In both cases, we establish the existence of infinitely many solutions, which either converge to zero or have larger and larger energies.
\vskip 0.5\baselineskip

\selectlanguage{francais}
% Text of abstract in French
\noindent{\bf R\'esum\'e} \vskip 0.5\baselineskip \noindent
On \'etudie le probl\`eme non lin\'eaire $\Delta u+a(x)u=\lambda g(x)f(u)$ dans $V\setminus V_0$,
$u=0$ sur $V_0$, o\`u $V$ est le joint de culasse de Sierpi\'nski, $V_0$ est sa fronti\`ere intrins\`eque, $\Delta$ d\'enote l'op\'erateur de Laplace au sens faible, $\lambda$ est un param\`etre positif et $f$ a un comportement oscillatoire autour de l'origine ou \`a l'infini. Dans les deux cas on \'etablit l'existence d'une infinit\'e de solutions, qui  ou bien convergent vers \`a z\'ero, ou bien ont des \'energies de plus en plus grandes.
\end{abstract}
\end{frontmatter}

% now the Version fran\c{c}aise abr\'eg\'ee, if it exists
\selectlanguage{francais}
\section*{Version fran\c{c}aise abr\'eg\'ee}
Soit $V$ le joint de culasse de Sierpi\'nski (ainsi appel\'e par Mandelbrot \cite{mandel}) et soit $V_0$ sa fronti\`ere intrins\`eque.
Le but de cette Note est d'\'etudier le probl\`eme de Dirichlet
$$
\left\{
\begin{array}{lll}
&\Delta u(x)+a(x)u(x)=\lambda g(x)f(u(x))&\qquad \mbox{si}\ x \in V\setminus V_0, \\
&u=0&\qquad\mbox{sur}\ V_0,\\
\end{array}
\right. \eqno(P)
$$
o\`u $\lambda$ est un param\`etre positif et les potentiels $a$ et $g$ satisfont les
conditions suivantes:

\smallskip \noindent
{$({\rm{h}}_1)$} $a\in L^1(V,\mu)$ et $a\leq0$ presque partout dans $V$, o\`u $\mu$ est la mesure normalis\'ee ${\log N}/{\log 2}$-dimensionnelle de Hausdorff sur $V$;\\
{$({\rm{h}}_2)$} $g\in C(V)$, $g\leq0$ et  la restriction de $g$ sur tout sous-ensemble ouvert de $V$ n'est pas identiquement nulle.

\smallskip
Dans le cas o\`u le terme non lin\'eaire $f$ a un comportement oscillatoire autour de l'origine, on
montre que si $\lambda>0$ est suffisament petit, alors le probl\`eme (P) admet une infinit\'e de solutions. Plus pr\'ecis\'ement, on a
 la propri\'et\'e suivante de multiplicit\'e.

\smallskip
\begin{theoreme} \label{th-1fr}
Soit $f:\RR\rightarrow [0,+\infty)$ une fonction continue.
On suppose que
$$
\displaystyle \liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{2}}<+\infty\,\,\,\,\,\,\,\ \textrm{and}\,\,\,\,\,\,\,\,\,\limsup_{\xi\rightarrow 0^{+}
}\frac{F(\xi)}{\xi^{2}}=+\infty,
$$
o\`u $F(t):=\di\int_0^tf(s)ds$.

 Alors, pour tout $$
\lambda\in\left(0,-\frac{1}{2(2N+3)^2\left(\di\int_Vg(x)d\mu\right) \displaystyle \liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{2}}}\right),
$$ il existe une suite $\{v_n\}$
 de solutions faibles du probl\`eme (P) qui converge vers z\'ero dans l'espace
$H^1_0(V)$.
\end{theoreme}

\smallskip
La preuve du Th\'eor\`eme \ref{th-1fr} repose  sur un r\'esultat de multiplicit\'e \'etablit par Bonanno et Molica Bisci (voir \cite[Theorem~2.1]{BMB}), qui \'etend le principe variationnel de Ricceri \cite{rice}. Avec des arguments similaires on peut traiter le cas suivant, qui correspond \`a un comportement oscillatoire de la nonlin\'earit\'e  \`a l'infini.

\smallskip
\begin{theoreme} \label{th-2fr}
Soit $f:\RR\rightarrow [0,+\infty)$ une fonction continue.
On suppose que
$$
\liminf_{\xi\rightarrow +\infty}\frac{F(\xi)}{\xi^{2}}<+\infty\,\,\,\,\,\,\,\ \textrm{and}\,\,\,\,\,\,\,\,\,\limsup_{\xi\rightarrow +\infty
}\frac{F(\xi)}{\xi^{2}}=+\infty.
$$

 Alors, pour tout $$
\lambda\in\left(0,-\frac{1}{2(2N+3)^2\left(\di\int_Vg(x)d\mu\right) \displaystyle \liminf_{\xi\rightarrow +\infty}\frac{F(\xi)}{\xi^{2}}}\right),
$$
\newpage
\noindent il existe une suite $\{v_n\}$
 de solutions faibles du probl\`eme (P) qui est non born\'ee dans l'espace
$H^1_0(V)$.
\end{theoreme}


\vspace{0.3cm}
$$\line(1,0){300}$$
\selectlanguage{english}

\vspace{1cm}
In this Note we are concerned with a class of nonlinear Dirichlet problems on the
Sierpi\'nski gasket $V$ in $\RR^{N-1}$, see Sierpi\'nski \cite{Sie}. We refer to
Falconer \cite{Fa} for the construction of the rigorous construction of this fractal set.
Let
$V_0$ denote the intrinsic
boundary of $V$. By Theorem 9.3 in Falconer
\cite{Fa}, the Hausdorff (fractal) dimension $d$ of $V$ satisfies  $d=\displaystyle{\log N}/{\log 2}$ and $0<{\mathcal H}^d(V)<\infty$,
where ${\mathcal H}^d$ is the $d$-dimensional Hausdorff measure on
$\RR^{N-1}$.

Let $\mu$ be the normalized restriction of ${\mathcal
H}^d$ to the subsets of $V$.  The following property
of $\mu$ will be useful in the sequel:
$$
\mu(B)>0,\qquad\mbox{ for every non-empty open subset $B$ of }V.
\eqno(1)$$
In other words, the support of $\mu$ coincides with $V$; see, for instance, Breckner, R\u adulescu and Varga \cite{BRaduV}.

Denote by
$C(V)$ the space of real-valued continuous functions on $V$ and vanishing on
$V_0$.
If $u:V\to\RR$ and  $m\in\NN$, we set
$$
W_m(u)=\left(\frac{N+2}{N}\right)^m\sum_{{x,y\in
V_m}:{|x-y|=2^{-m}}}(u(x)-u(y))^2,
$$
where $V_m$ is a suitable set associated to the fractal $V$ obtained recursively starting from $V_0$; see \cite{bmrcocv}.\par
Then $W_m(u)\leq W_{m+1}(u)$ for all $m$. Denote
$
W(u)=\lim_{m\to\infty} W_m(u)$.
Set $${\mathcal W}_m(u,v)=\left(\frac{N+2}{N}\right)^m\sum_{{x,y\in
V_m}:{|x-y|=2^{-m}}}(u(x)-u(y))(v(x)-v(y))$$
and
$${\mathcal W}(u,v)=\lim_{m\to\infty} {\mathcal W}_m(u,v).$$
Then $H_0^1(V):=\{u\in C_0(V)\mid W(u)<\infty\}$ is a Hilbert space with respect to the norm induced by the inner product ${\mathcal W}(u,v)$.


As pointed out by Falconer and Hu \cite{FaHu},  if $a\in L^1(V)$ and $a\leq 0$ in $V$ then
the norm
$
\|u\|_*:=\left({\mathcal W}(u,u)-\int_Va(x)u^2d\mu\right)^{1/2},
$
is equivalent with $\sqrt{W(u)}$ in $H^1_0(V)$.

 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 $L^2(V,\mu)$,
such that
$$-{\mathcal W}(u,v)=\int_V\Delta u\cdot vd\mu,\qquad \hbox{ for every }(u,v)\in Z\times H_0^1(V).$$
The operator $\Delta$ is called the weak Laplace operator on
$V$.


\medskip
In this Note we are concerned with the nonlinear problem
$$
\left\{
\begin{array}{lll}
&\Delta u(x)+a(x)u(x)=\lambda g(x)f(u(x))&\qquad \mbox{if}\ x \in V\setminus V_0, \\
&u=0&\qquad\mbox{on}\ V_0,\\
\end{array}
\right. \eqno(P)
$$
where $\lambda$ is a positive real parameter. We assume that $f:\RR\to\RR$ is a continuous function and that the variable potentials $a,\, g:V\to \RR$ satisfy the following conditions:\\
{$({\rm{h}}_1)$} $a\in L^1(V,\mu)$ and $a\leq0$ almost everywhere in $V$, where $\mu$ denotes the restriction to $V$ of the normalized ${\log N}/{\log 2}$-dimensional Hausdorff measure on $V$;\\
{$({\rm{h}}_2)$} $g\in C(V)$ with $g\leq0$ and such that the restriction of $g$ to every open subset of $V$ is not identically zero.

\smallskip
The first result of this Note establishes the following multiplicity property, provided that the nonlinearity has an oscillatory behaviour near the origin.

\begin{theorem}\label{Main1}
Let $f:\RR\rightarrow\RR$ be a non-negative continuous function.
Assume that
$$
\displaystyle \liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{2}}<+\infty\,\,\,\,\,\,\,\ \textrm{and}\,\,\,\,\,\,\,\,\,\limsup_{\xi\rightarrow 0^{+}
}\frac{F(\xi)}{\xi^{2}}=+\infty,
$$
where $F(t):=\di\int_0^tf(s)ds$.

 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 (P) such that
$\displaystyle\lim_{n\to\infty}\|v_n\|=\displaystyle\lim_{n\to\infty}\|v_n\|_\infty=0.$
\end{theorem}

{\it Proof of Theorem \ref{Main1}}.
We define the functionals
$\Phi,\Psi:H^1_0(V)\rightarrow\RR$ by
$$\Phi(u)=\frac12\|u\|^2-\frac12\int_Va(x)u^2(x)d\mu \ \ \ \mbox{and}\ \ \ \Psi(u)=-\int_Vg(x)F(u(x))d\mu.$$
 Fix $\lambda$ as in the conclusion. Then all critical points of the functional $I_\lambda:=\Phi-\lambda\Psi$ are weak solutions of problem (P).
We first observe that $I_\lambda\in C^{1}(H^1_0(V),\RR)$. Next, $\Phi$ is  coercive and, by Lemma 5.6 in Breckner, R\u adulescu and Varga \cite{BRaduV}, the functionals $\Phi$ and $\Psi$ are weakly sequentially lower
semi-continuous on $H^1_0(V)$.  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 \NN$.
Due to the compact embedding $(H_0^1(V),\|\cdot\|)\hookrightarrow (C_0(V),\|\cdot\|_\infty)
$ (see Fukushima and Shima \cite{FuSc}) we have
$$
\left\{v\in H^1_0(V) \mid \Phi(v)<r_n \right\}\subseteq \left\{v\in H^1_0(V) \mid \|v\|_\infty\leq c_n \right\}.
$$
Therefore
$$
\varphi(r_n)=\inf_{\Phi(u)<r_n}\frac{\displaystyle\sup_{\Phi(v)<
r_n}\displaystyle\int_V(-g(x))F(v(x))d\mu+\int_Vg(x)F(u(x))d\mu}{r_n-\Phi(u)}
$$
$$
\leq\frac{\displaystyle\sup_{\Phi(v)<
r_n}\displaystyle\int_V(-g(x))F(v(x))d\mu}{r_n}\leq -\left(\int_Vg(x)d\mu\right)\frac{\displaystyle\max_{|\xi|\leq c_n}F(\xi)}{r_n}
$$
$$
=
-\left(\di\int_Vg(x)d\mu\right)\frac{F(c_n)}{r_n}=-
2(2N+3)^2\left(\di\int_Vg(x)d\mu\right)\frac{F(c_n)}{c_n^{2}}.
$$
Thus, since
$ \liminf_{\xi\rightarrow 0^+}F(\xi)/\xi^{2}<+\infty,
$
we deduce that
$$
\displaystyle \delta\leq\liminf_{n\rightarrow
\infty}\varphi(r_n)\leq -2(2N+3)^2\left(\di\int_Vg(x)d\mu\right) \displaystyle \liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{2}}<+\infty.
$$
We prove in what follows that $0$, which is the unique global minimum of $\Phi$, is not a local minimum of the functional $I_\lambda$. Hence, fix a function $u\in H^1_0(V)$ 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 $V$. Define $h:\RR\to\RR$ by
$
h(t)=|\min\{t,1\}|$ for all $t\in\RR$.
Since $h(0)=0$ and $h$ is a Lipschitz function, we have  $v:=h\circ u\in H^1_0(V)$. 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^{+}
}F(\xi)/\xi^{2}=+\infty,
$
 there exists a sequence $\{\xi_n\}$
in $]0,\rho[$ such that $\lim_{n\to\infty}\xi_n=0$ and
$
\lim_{n\to\infty}F(\xi_n)/\xi_n^2=+\infty$.
Consider the sequence of functions $\{\xi_nv\}\subset H^1_0(V)$. We have
$$\begin{array}{ll}
\displaystyle I_\lambda(\xi_nv)&\displaystyle =\frac{\xi_n^2}{2}\|v\|^2-\frac{\xi_n^2}{2}\int_Va(x)v^2(x)d\mu \\ &\displaystyle +\lambda F(\xi_n)\int_Dg(x)d\mu
+\lambda\int_{V\setminus D}g(x)F(\xi_nv(x))d\mu.\end{array}
$$
 Taking into account that $F$ is positive in $]0,+\infty[$, from hypothesis $({\rm{h}}_2)$, the above equation yields
$$
I_\lambda(\xi_nv)\leq\frac{\xi_n^2}{2}\|v\|^2-\frac{\xi_n^2}{2}\int_Va(x)v^2(x)d\mu+\lambda F(\xi_n)\int_Dg(x)d\mu,\qquad\forall\; n\in \NN.
$$
\noindent Thus, for every $n\in\NN$,
$$\frac{I_\lambda (\xi_nv)}{\xi_n^2}\leq \frac12\|v\|^2-\frac12\int_Va(x)v^2(x)d\mu+\lambda\frac{F(\xi_n)}{\xi_n^2}\int_Dg(x)d\mu.$$
 Moreover, by $({\rm{h}}_2)$ and (1), we deduce that $\di\int_Dg(x)d\mu<0$.
Therefore
$$\lim_{n\to\infty}\frac{I_\lambda(\xi_nv)}{\xi_n^2}=-\infty,$$
  hence $I_\lambda(\xi_nv)<0$ for  $n$ sufficiently large. Since $I_\lambda(0)=0$, we conclude that $0$ is not a local minimum of $I_\lambda$. Moreover, since $\Phi$ has $0$ as unique global minimum, Theorem 2.1 in Bonanno and Molica Bisci \cite{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 It follows that $\lim_{n\rightarrow \infty}\|v_n\|=0$, which implies
$\lim_{n\to\infty}\|v_n\|_\infty=0$. This completes the proof of Theorem \ref{Main1}. \qed

\bigskip
With similar arguments we can prove the following result
in the case where the nonlinear term $f:\RR\to\RR$ has an oscillatory
behaviour at infinity.

\begin{theorem}\label{Main2}
Let $f:\RR\rightarrow\RR$ be a non-negative continuous function.
Assume that
$$
 \liminf_{\xi\rightarrow +\infty}\frac{F(\xi)}{\xi^{2}}<+\infty\,\,\,\,\,\,\,\ \textrm{and}\,\,\,\,\,\,\,\,\,\limsup_{\xi\rightarrow +\infty
}\frac{F(\xi)}{\xi^{2}}=+\infty.
$$
 Then, for every
$$
\lambda\in\left(0,-\frac{1}{2(2N+3)^2\left(\di\int_Vg(x)d\mu\right) \displaystyle \liminf_{\xi\rightarrow +\infty}\frac{F(\xi)}{\xi^{2}}}\right),
$$
there exists a sequence  of weak solutions
of problem (P) which is
unbounded in $H^1_0(V)$.
\end{theorem}




\smallskip
We refer to Bonanno, Molica Bisci, and R\u{a}dulescu \cite{bmrcocv} for detailed proofs, examples, and related results.


\section*{Acknowledgements}
V.~R\u adulescu acknowledges the support through  Grant CNCSIS PCCE--8/2010
{\it ``Sisteme diferen\c{t}iale \^{\i}n analiza neliniar\u{a} \c{s}i aplica\c{t}ii"}.



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\bibitem{bmrcocv} G. Bonanno, G. Molica Bisci and V. R\u adulescu,  Variational analysis for a nonlinear elliptic problem on the Sierpi\'nski gasket, {\it ESAIM: Control, Optimisation and Calculus of Variations}, in press (DOI: 10.1051/cocv/2011199).

\bibitem{BRaduV} B.E. Breckner, V. R\u{a}dulescu and Cs. Varga, Infinitely many solutions for the Dirichlet problem on the Sierpi\'nski gasket, {\it Analysis and Applications} {\bf 9} (2011), 235--248.

\bibitem{Fa}
K.J. Falconer, {\it Fractal Geometry: Mathematical Foundations and
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\bibitem{FaHu}
K.J. Falconer and J. Hu,  Nonlinear elliptical equations on
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\bibitem{FuSc}
M. Fukushima and T. Shima, On a spectral analysis for the Sierpi\'nski gasket, {\it  Potential Anal.} \textbf{1} (1992), 1--35.

\bibitem{mandel} B. Mandelbrot, {\it Les objets fractals: forme, hasard et dimension}, Flammarion, Paris, 1973.

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\bibitem{Sie} W. Sierpi\'nski, Sur une courbe dont tout point est un point de ramification, {\it Comptes Rendus (Paris)} {\bf 160} (1915), 302--305.


\end{thebibliography}

\end{document}
