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\title[Gradient-type systems...]
{Qualitative analysis of gradient-type systems with oscillatory nonlinearities 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]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{example}{Example}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{th}{Theorem}
\renewcommand{\theth}{\Alph{th}}
\newcommand{\erre}{{\mathbb R}}
\newcommand{\enne}{{\mathbb N}}
\def\essinf{\mathop {\rm ess\,inf}}
\def\di{\displaystyle}
\def\esssup{\mathop {\rm ess\,sup}}
\def\R{{\mathbb R}}
\def\RR{{\mathbb R}}
\def\N{{\mathbb N}}
\def\essinf{\mathop {\rm ess\,inf}}
\def\n{\noindent}
\def\phi{\varphi}
\def\div{\mathop{\rm div}}
\def\d{\mathop{\rm dist}}
\def\essinf{\mathop {\rm ess\,inf}}
\def\esssup{\mathop {\rm ess\,sup}}
\def\RR{{\mathbb R}}
\def\N{{\mathbb N}}
\def\essinf{\mathop {\rm ess\,inf}}
\def\n{\noindent}
\def\phi{\varphi}
\def\div{\mathop{\rm div}}
\def\meas{\mathop{\rm meas}}
\def\H{\mathop{H^1_0(\Omega)}}
\def\xseg{\bar{x}}
\def\useg{\bar{u}}
\def\d{\mathop{\rm dist}}

\def\weak{\rightharpoonup}
\def\ep{\varepsilon}
\def\half{{1\over2}}
\def\phi{\varphi}

\def\NN{{\mathbb N} }
\def\ZZ{{\mathbb Z} }
\def\supp{{\rm supp}}
\def\di{\displaystyle}
\def\ri{\rightarrow}
\def\intom{\int_\Omega}
\begin{document}


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\begin{abstract}
Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear data, the existence of a sequence of weak solutions for parametric quasilinear systems of gradient-type on the
Sierpi\'nski gasket is proved. Moreover, by adopting the same hypotheses on the potential and in presence of suitable small perturbations, the same conclusion is achieved. Our approach is based on
variational methods and on certain analytic and geometrical properties of the Sierpi\'nski fractal as, for instance, a compact embedding result due to Fukushima and Shima.
\end{abstract}

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\maketitle
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%INTRODUCTION%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
A particular interest has been given in the last few decays to the study of various nonlinear partial differential equations on fractal domains. For instance, many physical problems lead to nonlinear models involving
reaction-diffusion equations, problems on elastic fractal media or fluid flow through
fractal regions. We also point out relevant applications of the fractal theory to topology,
differential geometry, functional and harmonic
analysis, and probability theory. For recent advances in the theory of nonlinear
elliptic equations on fractals we refer to Barlow and Kigami \cite{barkig}, Bockelman and Strichartz \cite{bokstr}, Falconer
\cite{Fa99}, Falconer and Hu \cite{FaHu}, Hu \cite{Hu}, Hua and Zhenya \cite{HuaZh}, and Strichartz \cite{stri,stripaper}. The main tools used in some of these papers to prove the existence of nontrivial solutions or of multiple
solutions to nonlinear elliptic equations with zero Dirichlet boundary conditions defined
on fractals are certain minimax results (mountain pass or saddle point type theorems),
results from genus theory, and minimization procedures. A particular concern has been
devoted to PDEs on the Sierpi\'nski gasket.

 In the present paper we are interested in Dirichlet gradient-type systems of the form
\begin{equation}\tag{$S_{\lambda}$}\label{N}
\left\{
\begin{array}{l}
\Delta u_1(x)+a_1(x)u_1(x)=\lambda g(x)F_{u_1}(u_1(x),u_2(x))\quad x \in V\setminus V_0 \\
\Delta u_2(x)+a_2(x)u_2(x)=\lambda g(x)F_{u_2}(u_1(x),u_2(x))\quad x \in V\setminus V_0 \\
u_1|_{V_0}=u_2|_{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.\par We assume that
$F:\erre^{2}\rightarrow\erre$ is a $C^1$-function such that $F(0,0)=0$ and $F_{u_i}$ denotes the partial derivative of $F$
with respect to $u_i$. Finally, the variable potentials $a_1,a_2, g:V\to \R$ satisfy the following conditions:
\begin{itemize}
\item[{$({\rm{h}}_1)$}] $a_i\in L^1(V,\mu)$ and $a_i\leq0$ ($i=1,2$) 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
$\mu(V)=1$; see, for more details, the recent work \cite{BRV}.\par
 \indent The non-linear problem \eqref{N} is closely related to physical phenomena
such as reaction-diffusion problems and elastic properties of fractal media
and flow through fractal regions. There is an extensive theory for the study
of nonlinear elliptic equations
\eqref{N} on classical domains, that is, on open sets of $\erre^N$, for example, using
Sobolev spaces and Sobolev embedding theorems; see, for instance, the papers \cite{BdF,B2,B3} and also \cite{CdFM,dF2,h1}.
Many solvability conditions are given, such as the fibering method introduced by Pohozaev and the study of the Nehari manifold for some classes of quasilinear elliptic systems involving a pair of Laplacian operators; see \cite{BM,Za}.\par
A natural question
arises of how to establish an appropriate framework to cope with \eqref{N} on
fractal domains. Here we work on a specific fractal, the Sierpi\'nski gasket $V$ in $\R^{{N-1}}$
($N\geq 2$), which is typical of the more general class of post-critically finite
fractals.\par
 Over the years, the Sierpi\'nski 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.\par
 Moreover, this geometrical object
represents 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.\par
The importance of fractals is given by their utility in physics, chemistry or biology.
Moreover, the study of the Laplacian on fractals was originated in physics
literature, where so-called spectral decimation method was developed \cite{A,Ra,Ra2}. For completeness we recall that the Laplacian on the Sierpi\'nski gasket was first constructed as
the generator of a diffusion process; see \cite{gold,Ku}.\par
\indent Here we are interested on the existence of infinitely many solutions for a system of gradient-type (\ref{N}) by using variational methods. In our main results, just requiring an oscillating behaviour of the
term $F$ either at zero or at infinity, we prove, without
symmetry assumptions, that problem (\ref{N}) admits a sequence of
pairwise distinct weak solutions; see Theorems \ref{Main1} and
\ref{Main2} below.
 Our method strongly relies on the following critical points theorem, which is a more precise version of \cite[Theorem 2.5]{Ricc2}.
\begin{theorem}\cite[Theorem 2.1]{BMB}\label{BMB}
Let $E$ be a reflexive real Banach space, let $\Phi,\Psi:E\to\R$
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 $r>\inf_E
\Phi$, put
$$
\varphi(r):=\inf_{u\in \Phi^{-1}(]-\infty,r[)}\frac{\displaystyle\sup_{v\in \Phi^{-1}(]-\infty,r[)}\Psi(v)-\Phi(u)}{r-\Phi(u)},
$$
and
$$
\gamma:=\liminf_{r\to +\infty}\varphi(r),\qquad\delta:=\di\liminf_{r\to (\inf_E \Phi)^+}\varphi(r).
$$
Then, one has
\begin{itemize}
\item[\rm{(a)}] If $\gamma<+\infty$ then, for each
$\lambda\in\left]0,\displaystyle\frac{1}{\gamma}\right[$, the following
alternative holds:\\
either\\
{\rm $(\rm{a}_1)$}\ \ $J_\lambda:=\Phi-\lambda\Psi$ possesses a global minimum,\\
or\\
{\rm $(\rm{a}_2)$}\ \ there is a sequence $\{w_n\}$ of critical points
$($local minima$)$ of $J_\lambda$ such that $\displaystyle\lim_{n\to \infty}
\Phi(w_n)=+\infty$.
\item[\rm{(b)}] If $\delta<+\infty$ then, for each
$\lambda\in\left]0,\displaystyle\frac{1}{\delta}\right[$, the following
alternative holds:\\
either\\
{\rm $(\rm{b}_1)$}\ \ there is a global minimum of $\Phi$ which is a local minimum of $J_\lambda$,\\
or\\
{\rm $(\rm{b}_2)$}\ \ there is a sequence $\{w_n\}$ of pairwise distinct
critical points $($local minima$)$ of $J_\lambda$ which weakly
converges to a global minimum of $\Phi$, with $\displaystyle\lim_{n\to \infty}
\Phi(w_n)=\inf_{w\in E}\Phi(w)$.
\end{itemize}
\end{theorem}
\indent The above theorem 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 just assuming an appropriate oscillating behavior on the potential of the nonlinearity either at infinity or at zero; see Theorems \ref{Main1} and \ref{Main2}.\par

This method has been used
successfully to prove, in the context of certain Sobolev spaces, the
existence of infinitely many solutions for Dirichlet and Neumann equations; see \cite{BMB2,DMB0,DMB} and \cite{BoD,BoD2,BMB004,BMB04}. In the present paper we are able to show that the methods used
in \cite{BMB3} can be successfully adapted to prove the existence
of infinitely many (weak) solutions also for nonlinear elliptic systems of gradient-type on fractal domains.\par
 Many technical difficulties are overlapped in our approach by using suitable analytic properties arising from the geometry of the Sierpi\'nski gasket; see Remark \ref{SWcomparition}. For instance, a careful analysis of the normalized measure $\mu$, proved by Breckner, R\u{a}dulescu and Varga in \cite{BRaduV}, will be essential proving that, under certain hypotheses, the energy functional associated to \eqref{N} is unbounded from below; see the proof of Theorem \ref{Main1}.\par
In Corollaries \ref{infsol2} and \ref{infsol23} we discuss the case of a two-parametric Dirichlet system $(S_{\lambda,\mu})$, showing the existence of well-determined open interval of parameters $\lambda$ and $\mu$ for which problem $(S_{\lambda,\mu})$ admits infinitely many solutions; see \cite{DMB} for related topics.\par
 Further, existence results, analogous of Theorems \ref{Main1} and \ref{Main2}, for sign-changing potential can be obtained requiring an additional hypothesis on the nonlinearity either at infinity or at zero; see Remarks \ref{general} and \ref{general2}.\par
  Finally, we observe that if the data of our problems are sufficiently regular, every weak solution is a strong solution; see Remark \ref{strong} and Lemma 2.16 of Falconer and Hu \cite{FaHu}.\par
 \indent The plan of the paper is as follows. In Section 2 we recall some basic facts on suitable Sobolev spaces associated to the Sierpi\'nski gasket. Successively, Section 3 is devoted to the main theorems. A direct application is presented in the final section of this paper.\par
 We cite the very recent monograph \cite{KRV} as general reference for the basic notions used in the paper.
Finally, we denote by $\N$ the set of natural
numbers $\{0,\, 1,\, 2,\dots\}$ and by $|\cdot|$ the Euclidian norm on the
space $\R^N$.

\section{Preliminaries}
The Sierpi\'nski gasket has the origin in a paper by Sierpi\'nski \cite{Sie} and in a very simple manner
can be described as a subset of the plane obtained from an equilateral triangle by removing the open middle inscribed equilateral triangle
of $1/4$ 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
\indent Let $V$
be the \textit{Sierpi\'nski gasket} in $\R^{N-1}$ ($N\geq 2$) of intrinsic boundary $V_0$. For a direct construction and more details on this topics see the paper \cite{BRV}.\par
 Denote by
$C(V)$ the space of real-valued continuous functions on $V$ and by
$$C_0(V):=\{u\in C(V);\ u|_{V_0}=0\}.$$
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 very natural $m$, so we can put
\begin{equation}\label{defW}
W(u)=\lim_{m\to\infty} W_m(u).
\end{equation}
Define now
$$H_0^1(V):=\{u\in C_0(V);\ W(u)<\infty\}.$$
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
$${\mathcal W}_m(u,v)=\left(\frac{N+2}{N}\right)^m\sum_{\underset{|x-y|=2^{-m}}{x,y\in V_m}}(u(x)-u(y))(v(x)-v(y)).$$
Kigami's idea \cite{kigamipaper,K} is to approximate the fractal from
within by a sequence of finite graphs. The Laplace operator
on the fractal is then the renormalized limit
of graph Laplacians. This construction is described in what follows. We first define
$${\mathcal W}(u,v)=\lim_{m\to\infty} {\mathcal W}_m(u,v).$$

\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 just 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
$$
\|u\|_{-1}=\sup_{\underset{\|h\|=1}{h\in H^1_0(V)}} |\langle u,h\rangle |,
$$
\noindent where
$$
\langle v,h\rangle =\int_Vvhd\mu,
$$
\noindent $v\in L^2(V,\mu)$ and $h\in H^1_0(V)$. Then $H^{-1}(V)$ is a Hilbert space and the relation
$$
-{\mathcal W}(u,v)=\langle \Delta u,v\rangle ,\,\,\,\,\,\,\forall v\in H^1_0(V),
$$
uniquely defines a function $\Delta u\in H^{-1}(V)$ for every $u\in
H^1_0(V)$.\par

\indent Fix $\lambda>0$. We say that a function $(u_1,u_2)\in H^1_0(V)\times H^1_0(V)$ is called a
{\textit{weak solution} of} (\ref{N}) if
$$\sum_{i=1}^{2}\left[\left({\mathcal W}(u_i,v_i)-\int_Va_i(x)u_i(x)v_i(x)d\mu\right)+\lambda\int_Vg(x)F_{u_i}(u_1(x),u_2(x))v_i(x)d\mu\right]=0,$$
for every $(v_1,v_2)\in H^1_0(V)\times H^1_0(V)$.\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 $x\in V_m$.\par
 We say that $(u_1,u_2)\in C_0(V)\times C_0(V)$ is a {\it strong solution} of (\ref{N}) if $\Delta_su_1$ and $\Delta_su_2$ exist and are continuous for all $x\in V\setminus V_0,$ in addition to
$$
\left\{
\begin{array}{l}
\Delta_s u_1(x)+a_1(x)u_1(x)=\lambda g(x)F_{u_1}(u_1(x),u_2(x))\quad x \in V\setminus V_0 \\
\Delta_s u_2(x)+a_2(x)u_2(x)=\lambda g(x)F_{u_2}(u_1(x),u_2(x))\quad x \in V\setminus V_0 \\
u_1|_{V_0}=u_2|_{V_0}=0.\\
\end{array}
\right.
$$
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 $a_1,a_2\in C(V)$, arguing as in 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}
In this section we assume that
$F:\erre^{2}\rightarrow\erre$ is a $C^1$-function such that $F(0,0)=0$ and $F_{u_i}$ denotes the partial derivative of $F$
with respect to $u_i$. Moreover, the variable potentials $a_1,a_2, g:V\to \R$ satisfy the following conditions:
\begin{itemize}
\item[{$({\rm{h}}_1)$}] $a_i\in L^1(V,\mu)$ and $a_i\leq0$ ($i=1,2$) 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}
\indent For every $\xi>0$, set
$$
Q(\xi):=\Big\{(t_1,t_2)\in \erre^2:\sum_{i=1}^{2}|t_i|\leq \xi\Big\},
$$
and
$$
\erre^2_+=\{(t_1,t_2)\in \erre^2 : t_i\ge 0, \forall\;
i=1,2 \}.$$

 With the above notations, we have the following existence result in the case of nonlinearities oscillating at infinity.
\begin{theorem}\label{Main1}
Let $F$ be nonnegative on $\erre^2_+$. Further, assume that
$$
\displaystyle A_\infty:=\liminf_{\xi\rightarrow +\infty}\frac{\displaystyle \displaystyle\max_{(t_1,t_2)\in Q(\xi)}F(t_1,t_2)}{\xi^{2}}<+\infty,\eqno{({F}_{\infty}^{1})}
$$
and
$$
\limsup_{\begin{array}{ll}
    (t_1,t_2)\rightarrow \infty & \hbox{}  \\
    (t_1,t_2)\in \erre^2_+ & \hbox{}
  \end{array}}
  \frac{\displaystyle F(t_1,t_2)}{\displaystyle \sum_{i=1}^{2}t_i^{2}}=+\infty. \eqno{({F}_{\infty}^{2})}
$$
\noindent Then, for every
$$
\lambda\in\left]0,-\frac{1}{8(2N+3)^2\left(\di\int_Vg(x)d\mu\right)A_\infty}\right[,
$$
there exists an unbounded sequence of weak solutions
of problem \eqref{N}.
\end{theorem}

\begin{pf}
Let us define the functionals
$\Phi,\Psi:E\rightarrow\RR$ by
$$\Phi(u_1,u_2)=\frac12\sum_{i=1}^{2}\left(\|u_i\|^2-\int_Va_i(x)u_i(x)^2d\mu\right)$$
and
$$
\Psi(u_1,u_2)=-\int_Vg(x)F(u_1(x),u_2(x))d\mu,$$
 where, the product space $E:=H^1_0(V)\times H^1_0(V)$ is endowed by the norm
 $$
 \|(u_1,u_2)\|:=\sum_{i=1}^{2}\left(W(u_i)-\int_Va_i(x)u_i(x)^2d\mu\right)^{1/2},
 $$
 for every $(u_1,u_2)\in E$.
  Now, in order to achieved our goal, fix $\lambda$ as in the conclusion. Clearly, with the above notations, set $J_\lambda:=\Phi-\lambda\Psi$. First of all we observe that the functional $J_\lambda\in C^{1}(E,\R)$.\par
Further, fixing $(u_1,u_2)\in E$, one has
\begin{eqnarray*} J_\lambda'(u_1,u_2)(v_1,v_2)&=&\sum_{i=1}^{2}\left({\mathcal W}(u_i,v_i)-\int_Va_i(x)u_i(x)v_i(x)d\mu\right)\\
&+&\lambda\sum_{i=1}^{2}\int_Vg(x)F_{u_i}(u_1(x),u_2(x))v_i(x)d\mu,
\end{eqnarray*}
for each $(v_1,v_2)\in E$. In particular, $(u_1,u_2)\in E$ is a weak solution of problem \eqref{N}
if and only if $(u_1,u_2)$ is a critical point of $J_\lambda$.
 Clearly $\Phi$ is obviously coercive. Moreover, standard computations ensure that the functionals $\Phi$ and $\Psi$ are sequentially weakly lower
semi-continuous on $E$. Hence, we seek for
weak solutions of problem (\ref{N}) by applying part (\rm{a}) of Theorem \ref{BMB}.\par

Now let
 us verify that $\gamma<+\infty$. For our goal, let $\{c_n\}$ be a sequence (of positive numbers) such
that $\displaystyle \lim_{n\rightarrow \infty}c_n=+\infty$ and
$$
\lim_{n\rightarrow \infty}\displaystyle \frac{\displaystyle\max_{(t_1,t_2)\in Q(c_n)}F(t_1,t_2)}{c_n^{2}}=A_\infty.
$$
\noindent Put $\displaystyle r_n:=\frac{c_n^2}{8(2N+3)^2}$ for every $ n \in \N$.
Due to the compact embedding of $H^1_0(V)$ into $C_0(V)$, we have
$$
\left\{(v_1,v_2)\in E: \Phi(v_1,v_2)<r_n \right\}\subseteq \left\{(v_1,v_2)\in E: \sum_{i=1}^{2}|v_i(x)|\leq c_n\,\,\forall x\in V \right\}.
$$
Indeed, taking into account (\ref{embeddingconstant}), for every $v_i\in H^1_0(V)$ ($i=1,2$) one has
$$
\max_{x\in V}|v_i(x)|^{2}\leq (2N+3)^2 \|v_i\|^{2}.
$$
\noindent Consequently
$$
\max_{x\in V}\sum_{i=1}^{2}\frac{|v_i(x)|^{2}}{2}\leq (2N+3)^2 \Big(\sum_{i=1}^{2}\frac{\|v_i\|^{2}}{2}\Big)=(2N+3)^2\Phi(v_1,v_2).
$$
At this point, from the above inequality, if $\Phi(v_1,v_2)<r_n$, a direct computation ensures that
$\sum_{i=1}^{m}|v_i(x)|\leq c_n,$ for every $x\in V$.\par
Therefore, taking into account that $0_E\in \Phi^{-1}(]-\infty,r_n[)$, for every $n\in\enne$, since
$F(0,0)=0$, one has
\begin{eqnarray*}
\varphi(r_n)&=&\inf_{\sum_{i=1}^{2}\frac{\|u_i\|^{2}}{2}<r_n}\frac{\displaystyle\sup_{\sum_{i=1}^{2}\frac{\|v_i\|^{2}}{2}<r_n}\Psi(v_1,v_2)-\Psi(u_1,u_2)}{r_n-\displaystyle\big(\sum_{i=1}^{2}\frac{\|u_i\|^{2}}{2}\big)}\\
&\leq& \frac{\displaystyle\sup_{\sum_{i=1}^{2}\frac{\|v_i\|^{2}}{p_i}<r_n}\displaystyle\int_{V}-g(x)F(v_1(x),v_2(x))dx}{r_n}\\
&\leq& \displaystyle
\displaystyle
\displaystyle -8(2N+3)^2\left(\di\int_Vg(x)d\mu\right)\frac{\displaystyle\displaystyle\max_{(t_1,t_2)\in
Q(c_n)}F(t_1,t_2)}{c_n^{2}}.
\end{eqnarray*}
Thus, bearing in mind that
$$
\liminf_{\xi\rightarrow +\infty}\frac{\displaystyle \displaystyle\max_{(t_1,t_2)\in Q(\xi)}F(t_1,t_2)}{\xi^{2}}<+\infty,
$$
we deduce
$$
\displaystyle \gamma\leq\liminf_{n\rightarrow
\infty}\varphi(r_n)\leq -8(2N+3)^2\left(\di\int_Vg(x)d\mu\right) \liminf_{\xi\rightarrow +\infty}\frac{\displaystyle \displaystyle\max_{(t_1,t_2)\in Q(\xi)}F(t_1,t_2)}{\xi^{2}}<+\infty.
$$
\indent Let us verify that the
functional $J_\lambda$ is unbounded from below. For our goal, 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
 $$
 D:=\{x\in V;\ u(x)>1\},
 $$
 is a non-empty open (from the continuity of $u$) subset of $V$. Moreover, from relation (2.1) in \cite{BRV}, one has $\mu(D)>0$. Define $h\colon\R\to\R$ as follows
$$
h(t)=|\min\{t,1\}|,
$$
\noindent for every $t\in\erre$.\par
\noindent 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 $v:=h\circ u\in H_0^1(V)$. Moreover,
$v(x)=1$ for every $x\in D$, and $0\leq v(x)\leq1$ for every $x\in
V$.
From condition $({F}_{\infty}^{1})$ holds,
there exist $\{\xi_{1,n}\}$ and $\{\xi_{2,n}\}$
two positive real sequences such that
$$\displaystyle
\lim_{n\rightarrow
\infty}\sqrt{\sum_{i=1}^{2}\xi_{i,n}^2}=+\infty,$$ and
\begin{equation}\label{B}
\lim_{n\rightarrow \infty}\displaystyle \frac{\displaystyle F(\xi_{1,n},\xi_{2,n})}{\sum_{i=1}^{2}\xi_{i,n}^{2}}=+\infty.
\end{equation}
Consider the sequence of functions $\{(\xi_{1,n}v,\xi_{2,n}v)\}\subset E$. Thus
\begin{eqnarray}\label{jota}
J_\lambda(\xi_{1,n}v,\xi_{2,n}v)&=&\frac{\xi_{1,n}^2}{2}\|v\|^2-\frac{\xi_{1,n}^2}{2}\int_Va_1(x)v(x)^2d\mu\nonumber\\
&+& \frac{\xi_{2,n}^2}{2}\|v\|^2-\frac{\xi_{2,n}^2}{2}\int_Va_2(x)v(x)^2d\mu\\
&+& \lambda\int_Vg(x)F(\xi_{1,n}v(x),\xi_{2,n}v(x))d\mu.\nonumber
\end{eqnarray}
Now, since $F$ and $g$ are respectively nonnegative (on $\erre^2_+$) and nonpositive (on $V$), it follows that
\begin{eqnarray}\label{2i}
\int_Vg(x)F(\xi_{1,n}v(x),\xi_{2,n}v(x))d\mu&=&F(\xi_{1,n},\xi_{2,n})\int_Dg(x)d\mu\nonumber\\
&+&\int_{V\setminus D}g(x)F(\xi_{1,n}v(x),\xi_{2,n}v(x))d\mu\\
&\leq & F(\xi_{1,n},\xi_{2,n})\int_Dg(x)d\mu.\nonumber
\end{eqnarray}
Further, putting
$$
\vartheta(v):=\max\left\{\|v\|^2-\int_Va_1(x)v(x)^2d\mu, \|v\|^2-\int_Va_2(x)v(x)^2d\mu\right\},
$$
from \eqref{jota} and \eqref{2i} one has
\begin{eqnarray*}
J_\lambda(\xi_{1,n}v,\xi_{2,n}v)&\leq &\left(\frac{\sum_{i=1}^{2}\xi_{i,n}^2}{2}\right)\vartheta(v)+\lambda F(\xi_{1,n},\xi_{2,n})\int_Dg(x)d\mu,
\end{eqnarray*}
that means
\begin{eqnarray}\label{jota3}
\frac{J_\lambda(\xi_{1,n}v,\xi_{2,n}v)}{\sum_{i=1}^{2}\xi_{i,n}^2}&\leq &\frac{\vartheta(v)}{2}+\lambda \frac{F(\xi_{1,n},\xi_{2,n})}{\sum_{i=1}^{2}\xi_{i,n}^2}\int_Dg(x)d\mu.
\end{eqnarray}
At this point, since from \eqref{B} one has
$$
\lim_{n\rightarrow \infty}\frac{F(\xi_{1,n},\xi_{2,n})}{\sum_{i=1}^{2}\xi_{i,n}^2}\int_Dg(x)d\mu=-\infty,
$$
(note that $\displaystyle\int_Dg(x)d\mu<0$) inequality \eqref{jota3} implies that
$$
\lim_{n\rightarrow \infty}\frac{J_\lambda(\xi_{1,n}v,\xi_{2,n}v)}{\sum_{i=1}^{2}\xi_{i,n}^2}=-\infty.
$$
In conclusion
$$
\lim_{n\rightarrow \infty}J_\lambda(\xi_{1,n}v,\xi_{2,n}v)=-\infty,
$$
and the functional $J_\lambda$ is unbounded from below. Applying Theorem \ref{BMB} we deduce that the functional
$J_\lambda$ admits a sequence of critical points which is unbounded
in $E$. Hence, our claim is proved and the conclusion is achieved.
\end{pf}

\begin{remark}\label{SWcomparition}
\rm{It is worth important noting that the above statements and our method of proof are related to the corresponding ones in \cite{BMB3}. Clearly, the abstract framework introduced in the above mentioned paper is adaptable to our context by using the geometric and analytic properties of the Sierpi\'nski fractal as the Sobolev-type inequality
\begin{equation}\label{si}
\displaystyle\sup_{x,y\in V_*}\frac{|u(x)-u(y)|}{|x-y|^{\sigma}}\leq (2N+3)\sqrt{W(u)},
\end{equation}
\noindent where
$$
\sigma:=\displaystyle\frac{\log((N+2)/N)}{2\log 2},
$$
\noindent see, for more details, \cite[Lemma 2.4]{FaHu}.\par
\noindent We note that the estimate (\ref{si}) allows all $u: V_* \rightarrow \R$ of finite energy to have a
continuous extension to $V$. Moreover, through (\ref{si}) and by using the Ascoli-Arz\'{e}la theorem, the compact embedding (\ref{embedding}) is achieved.}
\end{remark}

\begin{remark}\label{general}\rm{
We explicitly observe that, exploiting the proof of Theorem
\ref{Main1}, one can see that the statements of our result are still true also for
sign-changing functions $F:\erre^2\rightarrow \R$ that satisfies assumptions $({F}_{\infty}^{1})$ and $({F}_{\infty}^{2})$ in addition to
$$
\liminf_{
    (t_1,t_2)\rightarrow \infty}\frac{F(t_1,t_2)}{\displaystyle\sum_{i=1}^{2}t_i^{2}}>-\infty.
$$
\noindent Indeed, if
$$
\liminf_{\xi\rightarrow +\infty}\frac{\displaystyle \displaystyle\max_{(t_1,t_2)\in Q(\xi)}F(t_1,t_2)}{\xi^{2}}<+\infty,
$$
one has that $\gamma<+\infty.$
On the other hand, consider the sequence of functions $\{(\xi_{1,n}v,\xi_{2,n}v)\}\subset E$ as in the proof of Theorem \ref{Main1}. From
$$
\liminf_{
    (t_1,t_2)\rightarrow \infty}\frac{F(t_1,t_2)}{\displaystyle\sum_{i=1}^{2}t_i^{2}}>-\infty,
$$
there exist $\varrho,k>0$ such that
\begin{equation}\label{mic5}
\frac{F(t_1,t_2)}{\displaystyle\sum_{i=1}^{2}t_i^{2}}\geq -k,\,\,\,\,\hbox{ if }\,\,\,\, \sum_{i=1}^{2}t_i^{2}>\varrho.
\end{equation}
Moreover, one has
\begin{eqnarray*}
J_\lambda(\xi_{1,n}v,\xi_{2,n}v)&=&\frac{\xi_{1,n}^2}{2}\|v\|^2-\frac{\xi_{1,n}^2}{2}\int_Va_1(x)v(x)^2d\mu+\frac{\xi_{2,n}^2}{2}\|v\|^2-\frac{\xi_{2,n}^2}{2}\int_Va_2(x)v(x)^2d\mu\\
&+& \lambda F(\xi_{1,n},\xi_{2,n})\int_Dg(x)d\mu+\lambda \int_{V\setminus D}g(x)F(\xi_{1,n}v(x),\xi_{2,n}v(x))d\mu,\nonumber
\end{eqnarray*}
\noindent for every $n\in \enne$.\par
\noindent Moreover
\begin{eqnarray*}
\int_{V\setminus D}g(x)F(\xi_{1,n}v(x),\xi_{2,n}v(x))d\mu
&=&\int_{G_{\varrho}\cap(V\setminus D)}g(x)F(\xi_{1,n}v(x),\xi_{2,n}v(x))d\mu\\
&+&\int_{G^{\varrho}\cap(V\setminus D)}g(x)F(\xi_{1,n}v(x),\xi_{2,n}v(x))d\mu,
\end{eqnarray*}
\noindent where
$$
G_\varrho:=\left\{x\in V:0\leq \left(\sum_{i=1}^{2}\xi_{i,n}^{2}\right)v(x)^2\leq\varrho\right\},
$$
and
$$
G^{\varrho}:=\left\{x\in V:\left(\sum_{i=1}^{2}\xi_{i,n}^{2}\right)v(x)^2>\varrho\right\}.
$$
\noindent Now, by using the mean value theorem, it follows that
\begin{equation}\label{glo}
\int_{G_{\varrho}\cap(V\setminus D)}g(x)F(\xi_{1,n}v(x),\xi_{2,n}v(x))d\mu\leq C,
 \end{equation}
 where
 $$
 C:=2\|g\|_\infty\max_{(t_1,t_2)\in [0,\sqrt{\varrho}]^2}|\nabla F(t_1,t_2)|\varrho.
 $$
 Then, relations (\ref{mic5}) and (\ref{glo}) yield
\begin{eqnarray*}
J_\lambda(\xi_{1,n}v,\xi_{2,n}v)&\leq &\left(\frac{\sum_{i=1}^{2}\xi_{i,n}^2}{2}\right)\vartheta(v)+\lambda F(\xi_{1,n},\xi_{2,n})\int_Dg(x)d\mu\\ &+& 2\lambda \|g\|_\infty\max_{(t_1,t_2)\in [0,\sqrt{\varrho}]^2}|\nabla F(t_1,t_2)|\varrho\\
&-&k\lambda\left(\sum_{i=1}^{2}\xi_{i,n}^{2}\right)\int_{V\setminus D}g(x)v^2(x)d\mu,
\end{eqnarray*}
\noindent for every $n\in\N$.\par
\noindent Thus, it is easy to see that the above inequality and our assumptions imply
$$
\lim_{n\rightarrow \infty}\frac{J_\lambda(\xi_{1,n}v,\xi_{2,n}v)}{\sum_{i=1}^{2}\xi_{i,n}^2}=-\infty.
$$
Then $J_\lambda$ is unbounded from below. The proof is attained from part \rm{(a)} of Theorem \ref{BMB}.
}
\end{remark}
 \indent Now, as a direct consequence of Theorem \ref{Main1}, we show that an appropriate oscillating behaviour of the nonlinear term $F$, even under certain small perturbations, ensures again the existence of infinitely many
solutions.\par
Denote by ${\mathfrak{F}}_V$ the set of functions $G:\erre^2\rightarrow\erre$
such that:
\begin{itemize}
\item[${g_1})$] $G$ is a $C^1$-function such that $G(0,0)=0$ and $G_{u_i}$ denotes the partial derivative of $G$
with respect to $u_i$;
\item[${g_2})$] $G(t_1,t_2)\geq 0$ for every $(t_1,t_2)\in \erre^2_+$.
\end{itemize}
\smallskip
\indent We have the following existence result.
\begin{corollary}\label{infsol2}
Under assumptions of Theorem \ref{Main1}, for each
$$
\lambda\in\left]0,-\frac{1}{8(2N+3)^2\left(\di\int_Vg(x)d\mu\right)A_\infty}\right[,
$$
for every $G\in{\mathfrak{F}}_V$ satisfying
\begin{itemize}
\item[${g_\infty})$] $G_\infty:=\displaystyle \lim_{\xi \to +\infty}\frac{\displaystyle \displaystyle\max_{(t_1,t_2)\in Q(\xi)}G(t_1,t_2)}{\xi^{2}}<+\infty$$,$
\end{itemize}
and for every $\mu\in[0,\mu_{G,\lambda}[$, where
$$
\mu_{G,\lambda}:=-\frac{1}{8(2N+3)^2\left(\di\int_Vg(x)d\mu\right)G_\infty }\left(1+8{\lambda}(2N+3)^2\left(\di\int_Vg(x)d\mu\right)A_\infty\right),
$$
\noindent the problem $($namely $(S_{\lambda,\mu})$$)$ given by
\begin{equation*}
\left\{
\begin{array}{l}
\Delta u_1(x)+a_1(x)u_1(x)=g(x)(\lambda F_{u_1}(u_1(x),u_2(x))+\mu G_{u_1}(u_1(x),u_2(x)))\quad x \in V\setminus V_0 \\
\Delta u_2(x)+a_2(x)u_2(x)=g(x)(\lambda F_{u_2}(u_1(x),u_2(x))+\mu G_{u_1}(u_1(x),u_2(x)))\quad x \in V\setminus V_0 \\
u_1|_{V_0}=u_2|_{V_0}=0,\\
\end{array}
\right.
\end{equation*}
\noindent admits a sequence of weak solutions which is
unbounded in $H_0^1(V)\times H_0^1(V)$.
\end{corollary}

\begin{pf}
Fix
 $
\overline{\lambda}\in\left]0,\lambda_2\right[,
$
where
$$
\lambda_2:=-\frac{1}{8(2N+3)^2\left(\di\int_Vg(x)d\mu\right)A_\infty},
$$
\noindent and let $G\in{\mathfrak{F}}_V$ be a function satisfies hypothesis $g_\infty)$. In the non-perturbed case, that is $\mu=0$, the assertion is trivial.
Otherwise, one has
$$
\mu_{G,\overline{\lambda}}:=-\frac{1}{8(2N+3)^2\left(\di\int_Vg(x)d\mu\right)G_\infty }\left(1+8{\overline{\lambda}}(2N+3)^2\left(\di\int_Vg(x)d\mu\right)A_\infty\right)>0.
$$
\noindent Take $0<\overline{\mu}<\mu_{G,\overline{\lambda}}$ and put
$$
\eta_2:=\displaystyle -\frac{1}{8(2N+3)^2\left(\di\int_Vg(x)d\mu\right)\left(\displaystyle A_\infty+\frac{\overline{\mu}}{\overline{\lambda}}G_\infty\right)}.
$$
\noindent If $G_\infty=0$ clearly one has $\eta_2=\lambda_2$ and
$$
\overline{\lambda}\in \Lambda^{\star}:=]0,\eta_2[.
$$
\noindent Otherwise, if $G_\infty\neq 0$, from $\overline{\mu}<\mu_{g,\overline{\lambda}}$, it follows that
$$
-8(2N+3)^2\left(\di\int_Vg(x)d\mu\right)\left(A_\infty\overline{\lambda}+G_\infty\overline{\mu}\right)<1,
$$
that means
$$
\overline{\lambda}<-\displaystyle \frac{1}{8(2N+3)^2\left(\di\int_Vg(x)d\mu\right)\displaystyle \left(A_\infty+\frac{\overline{\mu}}{\overline{\lambda}}G_\infty\right)}=\eta_2.
$$
\noindent Hence, also in this case, one has
$$
\overline{\lambda}\in \Lambda^{\star}:=]0,\eta_2[.
$$
\noindent Now, put
 $$
H(t_1,t_2):=F(t_1,t_2)+\dfrac{\overline{\mu}}{\overline{\lambda}}G(t_1,t_2),
$$
\noindent for every $t_1,t_2\in\erre$.\par
\noindent Taking into account that the potential $G$ is a nonnegative function, we obtain
$$
 \limsup_{\begin{array}{ll}
    (t_1,t_2)\rightarrow \infty & \hbox{}  \\
    (t_1,t_2)\in \erre^2_+ & \hbox{}
  \end{array}}
  \frac{\displaystyle H(t_1,t_2)}{\displaystyle \sum_{i=1}^{2}t_i^{2}}\geq \limsup_{\begin{array}{ll}
    (t_1,t_2)\rightarrow \infty & \hbox{}  \\
    (t_1,t_2)\in \erre^2_+ & \hbox{}
  \end{array}}
  \frac{\displaystyle F(t_1,t_2)}{\displaystyle \sum_{i=1}^{2}t_i^{2}}=+\infty,
  $$
then
$$
\limsup_{\begin{array}{ll}
    (t_1,t_2)\rightarrow \infty & \hbox{}  \\
    (t_1,t_2)\in \erre^2_+ & \hbox{}
  \end{array}}
  \frac{\displaystyle H(t_1,t_2)}{\displaystyle \sum_{i=1}^{2}t_i^{2}}=+\infty.
$$
\noindent Moreover, since
$$\frac{\displaystyle \displaystyle\max_{(t_1,t_2)\in Q(\xi)}H(t_1,t_2)}{\xi^{2}}\leq \frac{\displaystyle \displaystyle\max_{(t_1,t_2)\in Q(\xi)}F(t_1,t_2)}{\xi^{2}}
+\frac{\overline{\mu}}{\overline{\lambda}}\frac{\displaystyle \displaystyle\max_{(t_1,t_2)\in Q(\xi)}G(t_1,t_2)}{\xi^{2}},
$$
and taking into account hypothesis $g_\infty)$, it follows that
$$
 \liminf_{\xi\rightarrow
+\infty}\frac{\displaystyle \displaystyle\max_{(t_1,t_2)\in Q(\xi)}H(t_1,t_2)}{\xi^{2}}\leq \liminf_{\xi\rightarrow
+\infty}\frac{\displaystyle \displaystyle\max_{(t_1,t_2)\in Q(\xi)}F(t_1,t_2)}{\xi^{2}}+\frac{\overline{\mu}}{\overline{\lambda}}G_\infty<+\infty.
$$
\noindent Hence, owing to
$$
\overline{\lambda}\in\Lambda^{\star}\subseteq \left]0,-\frac{1}{8(2N+3)^2\left(\di\int_Vg(x)d\mu\right)H_\infty}\right[,
$$
where
$$
H_\infty:=\liminf_{\xi\rightarrow +\infty}\frac{\displaystyle \displaystyle\max_{(t_1,t_2)\in Q(\xi)}H(t_1,t_2)}{\xi^{2}},
$$
from Theorem \ref{Main1}, there exists an unbounded sequence of infinitely many solutions for problem $(S_{\overline{\lambda},\overline{\mu}})$.
The proof is complete.
\end{pf}

By the same method, applying part (\rm{b}) instead of part \rm{(a)}
of Theorem \ref{BMB}, one can prove the analogous of Theorem \ref{Main1}
in presence of a smooth nonlinear term $F\colon\R^2\to\R$ with a suitable
oscillating behaviour at zero.

\begin{theorem}\label{Main2}
Let $F$ be nonnegative on $\erre^2_+$. Further, assume that
$$
\displaystyle A_0:=\liminf_{\xi\rightarrow 0^+}\frac{\displaystyle \displaystyle\max_{(t_1,t_2)\in Q(\xi)}F(t_1,t_2)}{\xi^{2}}<+\infty,\eqno{({F}_{0}^{1})}
$$
and
$$
\limsup_{\begin{array}{ll}
    (t_1,t_2)\rightarrow (0,0) & \hbox{}  \\
    (t_1,t_2)\in \erre^2_+ & \hbox{}
  \end{array}}
  \frac{\displaystyle F(t_1,t_2)}{\displaystyle \sum_{i=1}^{2}t_i^{2}}=+\infty. \eqno{({F}_{0}^{2})}
$$
\noindent Then, for every
$$
\lambda\in\left]0,-\frac{1}{8(2N+3)^2\left(\di\int_Vg(x)d\mu\right)A_\infty}\right[,
$$
there exists a sequence $\{(u_{1,n},u_{2,n})\}\subset H_0^1(V)\times H_0^1(V)$ of pairwise distinct weak solutions
of problem \eqref{N} such that
$\displaystyle\lim_{n\to\infty}\|(u_{1,n},u_{2,n})\|=0.$
\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
$$
\liminf_{\xi\rightarrow 0^+}\frac{\displaystyle \displaystyle\max_{(t_1,t_2)\in Q(\xi)}F(t_1,t_2)}{\xi^{2}}<+\infty,
$$
\noindent by direct computations, it follows that $\delta:=\di\liminf_{r\to 0^+}\varphi(r)<+\infty$. On the other hand, by
$$
\limsup_{\begin{array}{ll}
    (t_1,t_2)\rightarrow (0,0) & \hbox{}  \\
    (t_1,t_2)\in \erre^2_+ & \hbox{}
  \end{array}}
  \frac{\displaystyle F(t_1,t_2)}{\displaystyle \sum_{i=1}^{2}t_i^{2}}=+\infty,
$$
there exist
two positive real sequences, namely $\{\xi_{1,n}\}$ and $\{\xi_{2,n}\}$, such that
$$\displaystyle
\lim_{n\rightarrow
\infty}\sqrt{\sum_{i=1}^{2}\xi_{i,n}^2}=0,$$ and
$$
\lim_{n\rightarrow \infty}\displaystyle \frac{\displaystyle F(\xi_{1,n},\xi_{2,n})}{\sum_{i=1}^{2}\xi_{i,n}^{2}}=+\infty.
$$

\noindent Now, consider the sequence of functions $\{(\xi_{1,n}v,\xi_{2,n}v)\}\subset E$. Arguing as in Theorem \ref{Main1}, we obtain
$$
\lim_{n\rightarrow \infty}\frac{J_\lambda(\xi_{1,n}v,\xi_{2,n}v)}{\sum_{i=1}^{2}\xi_{i,n}^2}=-\infty.
$$
\noindent Thus $J_\lambda(\xi_{1,n}v,\xi_{2,n}v)<0$ for every $n$ sufficiently large. Since $J_\lambda(0_E)=\Phi(0_E)-\lambda \Psi(0_E)=0$, the last inequality means that $0_E$ is not a local minimum of $J_\lambda$. Moreover, since $\Phi$ has $0_E$ as unique global minimum, Theorem \ref{BMB} ensures the existence of a sequence $\{(u_{1,n},u_{2,n})\}\subset E$ of pairwise distinct critical points of the functional $J_\lambda$, such that
  $$\lim_{n\rightarrow\infty}\sum_{i=1}^{2}\left(\|u_{i,n}\|^2-\int_Va_i(x)u_{i,n}(x)^2d\mu\right)=0.$$
\noindent Hence, one has that $\lim_{n\rightarrow \infty}\|(u_{1,n},u_{2,n})\|=0$. The proof is complete.
\end{pf}

\begin{remark}\label{general2}\rm{
Theorem \ref{Main2} also holds for
sign-changing functions $F:\erre^2\rightarrow \R$ that satisfies assumptions $({F}_{0}^{1})$ and $({F}_{0}^{2})$ in addition to
$$\liminf_{\begin{array}{ll}
    (t_1,t_2)\rightarrow (0,0) & \hbox{}  \\
    (t_1,t_2)\in \erre^2_+ & \hbox{}
  \end{array}}\frac{F(t_1,t_2)}{\displaystyle\sum_{i=1}^{2}t_i^{2}}>-\infty.
$$

}
\end{remark}

The next result can be view as an analogous of Corollary \ref{infsol2}.
\begin{corollary}\label{infsol23}
Under assumptions of Theorem \ref{Main2}, for each
$$
\lambda\in\left]0,-\frac{1}{8(2N+3)^2\left(\di\int_Vg(x)d\mu\right)A_0}\right[,
$$
for every $G\in{\mathfrak{F}}_V$ satisfying
\begin{itemize}
\item[${g_0})$] $G_0:=\displaystyle \lim_{\xi \to 0^+}\frac{\displaystyle \displaystyle\max_{(t_1,t_2)\in Q(\xi)}G(t_1,t_2)}{\xi^{2}}<+\infty$$,$
\end{itemize}
and for every $\mu\in[0,\mu_{G,\lambda}^{\star}[$, where
$$
\mu_{G,\lambda}^{\star}:=-\frac{1}{8(2N+3)^2\left(\di\int_Vg(x)d\mu\right)G_0}\left(1+8{\lambda}(2N+3)^2\left(\di\int_Vg(x)d\mu\right)A_0\right),
$$
\noindent the problem $($namely $(S_{\lambda,\mu})$$)$ given by
\begin{equation*}
\left\{
\begin{array}{l}
\Delta u_1(x)+a_1(x)u_1(x)=g(x)(\lambda F_{u_1}(u_1(x),u_2(x))+\mu G_{u_1}(u_1(x),u_2(x)))\quad x \in V\setminus V_0 \\
\Delta u_2(x)+a_2(x)u_2(x)=g(x)(\lambda F_{u_2}(u_1(x),u_2(x))+\mu G_{u_1}(u_1(x),u_2(x)))\quad x \in V\setminus V_0 \\
u_1|_{V_0}=u_2|_{V_0}=0,\\
\end{array}
\right.
\end{equation*}
admits a sequence $\{(u_{1,n},u_{2,n})\}\subset H_0^1(V)\times H_0^1(V)$ of pairwise distinct weak solutions
such that
$\displaystyle\lim_{n\to\infty}\|(u_{1,n},u_{2,n})\|=0.$
\end{corollary}

We want to conclude with this explicative application.

\begin{example}\label{Ex.2}
\rm{ \noindent Consider the
increasing sequence of positive real numbers given by
$$a_1:=2,
\,\,\,\,\,\,\,\,\,\,\, a_{n+1}:=n!a_n^2+2,$$ \noindent
for every $n\geq 1$. Define the $C^1$-function $F:\erre^2\rightarrow\erre$
as follows
$$
F(t_1,t_2):=\left\{
\begin{array}{l}
a_{n+1}^4 e^{-\frac{1}{1-[(t_1-a_{n+1})^2+(t_2-a_{n+1})^2]}+1}   \,\,\,\,\,\,\,\,\,\, \textrm{if}\,\,\, \displaystyle (t_1,t_2)\in \bigcup_{n\geq 1}{B}((a_{n+1},a_{n+1}),1) \\
0 \qquad \qquad\qquad\qquad\qquad \qquad\qquad\,{\rm otherwise,}
\end{array}
\right.
$$
\noindent where ${B}((a_{n+1},a_{n+1}),1)$ denote the open unit ball
of center $(a_{n+1},a_{n+1})$.\par \noindent By definition $F$ is
non-negative and $F(0,0)=0$. We will denote by $F_{u_1}$ and $F_{u_2}$ respectively
the partial derivative of $F$ respect to $u_1$ and $u_2$. Now, for
every $n\in\enne$, the restriction
$F(t_1,t_2)|_{{B}((a_{n+1},a_{n+1}),1)}$ attains its maximum at
$(a_{n+1},a_{n+1})$ and one has $F(a_{n+1},a_{n+1})=a_{n+1}^4$.
Clearly $$\limsup_{{\begin{array}{ll}
    (t_1,t_2)\rightarrow \infty & \hbox{}  \\
    (t_1,t_2)\in \erre^2_+ & \hbox{}
  \end{array}}}\frac{F(t_1,t_2)}{\displaystyle\sum_{i=1}^{2}t_i^{2}}=+\infty,$$
owing to
$$
\lim_{n\rightarrow \infty
}\frac{F(a_{n+1},a_{n+1})}{2a_{n+1}^2}=+\infty.
$$
\noindent On the other hand, by setting $y_n=a_{n+1}-1$ for every $n \in \enne$, one has
$$\displaystyle\max_{(t_1,t_2)\in Q(y_n)}F(t_1,t_2)=a_n^4,\,\,\,\,\,\,\,\forall\; n\in \enne.$$
Then
$$
\lim_{n \to \infty}\frac{\displaystyle\max_{(t_1,t_2)\in
Q(y_n)}F(t_1,t_2)}{(a_{n+1}-1)^2}=0,
$$
and
$$
\displaystyle \liminf_{\xi\rightarrow
+\infty}\frac{\displaystyle\max_{(t_1,t_2)\in Q(\xi)}F(t_1,t_2)}{\xi^2}=
0.
$$
\noindent The previous computations ensure that
all the hypotheses of Theorem \ref{Main1} are satisfied. Then, for every
$\lambda\in]0,+\infty[$, the following problem
\begin{equation}\tag{$S_{\lambda}'$}\label{N243}
\left\{
\begin{array}{l}
-\Delta u_1(x)=\lambda F_{u_1}(u_1(x),u_2(x))\quad x \in V\setminus V_0, \\
-\Delta u_2(x)=\lambda F_{u_2}(u_1(x),u_2(x))\quad x \in V\setminus V_0, \\
u_1|_{V_0}=u_2|_{V_0}=0,\\
\end{array}
\right.
\end{equation}

\noindent admits a sequence of (strong) solutions which is unbounded in
$H_0^1(V)\times H_0^1(V)$. }
\end{example}
\begin{remark}\label{gin}\rm{Fix an integer $m\geq 1$. Moreover,
let $F:\erre^{m}\rightarrow\erre$ be a $C^1$-function such that $F(0,...,0)=0$ and $F_{u_i}$ denotes the partial derivative of $F$
with respect to $u_i$. We explicitly observe that our results hold true also for Dirichlet systems of the form
\begin{equation*}
\left\{
\begin{array}{l}
\Delta u_k(x)+a_k(x)u_k(x)=\lambda g(x)F_{u_k}(u_1(x),...,u_m(x))\quad x \in V\setminus V_0;\,\,\,(1\leq k\leq m) \\
u_1|_{V_0}=...=u_k|_{V_0}=0,\\
\end{array}
\right.
\end{equation*}
assuming that the functions $a_k,g$ satisfy conditions like $({\rm{h}}_1)$ and $({\rm{h}}_2)$.
For instance, for every $\xi>0$, set
$$
Q_m(\xi):=\Big\{(t_1,...,t_m)\in \erre^m:\sum_{i=1}^{m}|t_i|\leq \xi\Big\},
$$
and
$$
\erre^m_+=\{(t_1,...,t_m)\in \erre^m : t_i\ge 0, \forall\;
i=1,...,m\}.$$
Requiring that
$$
\displaystyle A_\infty:=\liminf_{\xi\rightarrow +\infty}\frac{\displaystyle \displaystyle\max_{(t_1,...,t_m)\in Q_m(\xi)}F(t_1,...,t_m)}{\xi^{2}}<+\infty,
$$
and
$$
\limsup_{\begin{array}{ll}
    (t_1,...,t_m)\rightarrow \infty & \hbox{}  \\
    (t_1,...,t_m)\in \erre^m_+ & \hbox{}
  \end{array}}
  \frac{\displaystyle F(t_1,...,t_m)}{\displaystyle \sum_{i=1}^{m}t_i^{2}}=+\infty,
$$
\noindent for every
$$
\lambda\in\left]0,-\frac{1}{2m^2(2N+3)^2\left(\di\int_Vg(x)d\mu\right)A_\infty}\right[,
$$
there exists an unbounded sequence of weak solutions (in $\prod_{k=1}^{m}H^{1}_0(V)$)
 of the above problem. From the above remark one clearly has that Theorem \ref{Main1} extends \cite[Theorem 3.2]{BMB5} in the gradient-type setting.}
\end{remark}
Finally, for completeness, among the contributions to the theory of nonlinear elliptic equations on
fractals we mention \cite{BRV,Fa99,FaHu,Hu,HuaZh,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.\par

\medskip
 {\bf Acknowledgements.}  V.~R\u adulescu acknowledges the support through  Grant CNCS PCE--47/2011
{\it ``Qualitative and numerical analysis of nonlinear problems on fractals"} (PN-II-ID-PCE-2011-3-0195).

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\end{document} 