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\title[Elliptic problems on fractal sets]
{A characterization for elliptic problems\\ on fractal sets}


\author{Giovanni Molica Bisci and Vicen\c{t}iu D. R\u{a}dulescu}
\address[G. Molica Bisci]{Department P.A.U., University of Reggio Calabria, 89124 - Reggio Calabria, Italy} \email{gmolica@@unirc.it}
\address[V. R\u{a}dulescu]{Department of Mathematics, Faculty of Science, King Abdulaziz University,
Jeddah, Saudi Arabia} \email{vicentiu.radulescu@@math.cnrs.fr}

\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}
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\def\di{\displaystyle}
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\def\weak{\rightharpoonup}
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\def\phi{\varphi}

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\def\ZZ{{\mathbb Z} }
\def\supp{{\rm supp}}
\def\di{\displaystyle}
\def\ri{\rightarrow}
\def\intom{\int_\Omega}
\def\Int{{\rm int}}


\begin{document}


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\begin{abstract}
In this paper we prove a characterization theorem on the existence of one non-zero strong solution for elliptic equations defined on
the Sierpi\'nski gasket. More generally, the validity of our result can be checked studying elliptic equations defined on self-similar fractal
domains whose spectral dimension $\nu\in (0,2)$. Our theorem can be viewed as an elliptic version on fractal domains of a recent contribution obtained in \cite{ricceri} for a two-point boundary value problem.
\end{abstract}

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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%INTRODUCTION%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

A great deal effort in recent years has gone into investigating PDEs on fractals (see, for instance, \cite{Fa99, ZHe,HuaZh,stripaper} and the excellent monograph \cite{Fa}). A major difficulty is how to define differential operators on non-smooth sets. Analysis on fractal sets has been made possible by the definition of operators that play the role of the Laplacian.

Originally defined as a by-product of the construction of the analog of Brownian motion \cite{barlow}, these Laplace-type operators
have been made given by direct limit-of-difference-quotient definitions in the papers by Kigami \cite{kigami1,kigami2,kigami3,kigami4}, for a class of self-similar fractals that includes
the {\it Sierpi\'nski gasket}. In this way, elliptic equations have been studied by using a suitable energy functional defined on an appropriate Hilbert space (see \cite{BRaduV, BRV, FaHu, Hu}).\par

\indent Motivated by this large interest in the current literature, the purpose of this paper is to prove a characterization result on the existence of non-negative and non-zero strong solutions for the following Dirichlet problem
\begin{equation}\tag{$S_{\lambda, \alpha}^{f}$}\label{N}
\left\{
\begin{array}{l}
\Delta u(x)=\lambda \alpha(x)f(u(x))\quad x \in V\setminus V_0, \\
u|_{V_0}=0,\\
\end{array}
\right.
\end{equation}
where $V$ stands for the Sierpi\'nski gasket in $(\RR^{N-1},|\cdot|)$, $N\geq 2$, $V_0$ is its intrinsic boundary (consisting of its $N$ corners), $\Delta$ denotes the weak Laplacian on $V$ and $\lambda$ is a positive real parameter. We assume that

 \begin{itemize}
\item[{$({\rm{h}}_f)$}] \textit{$f:[0,+\infty)\to[0,+\infty)$ is a continuous function with $f(0)=0$ and such that, for some $a>0$, the map $h_F:[0,+\infty)\to[0,+\infty)$ defined by
$$
h_F(\xi):=\frac{\displaystyle F(\xi)}{\xi^2},
$$
is non-increasing in the real interval $(0,a]$, where  $$F(\xi):=\displaystyle\int_0^{\xi}f(t)dt,$$
for each $\xi\in [0,+\infty)$.}
\end{itemize}

We assume that the variable potential $\alpha:V\to \R$ satisfies the following hypothesis:
\begin{itemize}
\item[{$({\rm{h}}_\alpha)$}] \textit{$\alpha\in C(V)$ with $\alpha(x)<0$, for every $x\in V$.}
\end{itemize}
\smallskip

\indent The main result of this paper is the following.\par

\begin{theorem}\label{BMB}
Assume that hypotheses $({\rm{h}}_f)$ and $({\rm{h}}_\alpha)$ are fulfilled. Then, the following properties are equivalent$:$
\begin{itemize}
\item[$(\rm i_1)$] $h_F$ is not constant in $(0,b]$ for each $b>0$$;$
\item[$(\rm i_2)$] for each $r>0$ there exists an open interval $I\subseteq (0,+\infty)$ such that, for every $\lambda\in I$, problem \eqref{N} has a strong non-negative and non-zero solution, whose norm in $H_0^{1}(V)$ is less than $r$.
\end{itemize}
\end{theorem}

The above characterization can be regarded as an elliptic version, for some classes of fractal sets, of a very recent result obtained by Ricceri for a two-point boundary value problem (see \cite[Theorem 1]{ricceri}).\par

The extension of the cited result to \eqref{N} is not trivial and requires to overcome some difficulties which arise in this new geometrical context. In particular, some analytical properties on the Hilbert space $H^1_0(V)$ and the distribution of the spectrum of the corresponding linear problem defined on fractal sets, need a special care. {More precisely, in our setting, a key ingredient is the validity of the following Morrey-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 and $V_*$ is defined in the next section (see \cite[Lemma 2.4]{FaHu} for details).} Inequality \eqref{si} and the Arzela-Ascoli theorem yield that the embedding
 \begin{equation}\label{si2}
H_0^1(V)\hookrightarrow C_0(V)
\end{equation}
 is compact (see \cite{FuSc}). This fact will be crucial in our approach.\par
 Taking into account the results contained in \cite{Fa99}, our method adopted here can be useful for studying the existence of weak solutions for elliptic equations defined on self-similar sets, whose spectral dimension $\nu\in (0,2)$. In such a case, the Laplacian may be defined via a suitable Dirichlet
form, following the variational fractal approach developed by Mosco in \cite{Mosco}. An open and more delicate problem is to attack the case $\nu\geq 2$ in which the compact embedding \eqref{si2} is false.

  We emphasize that, as suggested by  Ricceri, a possible extension of his result to the elliptic case require a more sophisticated and delicate analysis also for equations involving the classical Laplacian and defined on bounded Euclidean domains.

 This paper is organized as follows. In Section 2 we recall the geometrical construction of the Sierpi\'nski gasket and our variational framework. Successively, Section 3 is devoted to the proof of the main theorem.

 We refer to the recent book by Ciarlet \cite{ciarlet} as general reference for the basic notions used in the present paper.

\section{Preliminaries and Abstract Result}

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
$$S_i(x)=\frac12\,x+\frac12\,p_i\,.$$
\indent Let
${\mathcal S}:=\{S_1,\dots, S_N\}$ and denote by $L\colon{\mathcal
P}(\R^{N-1})\to{\mathcal P}(\R^{N-1})$ the map assigning to a subset
$A$ of $\R^{N-1}$ the set
$$L(A)=\bigcup_{i=1}^NS_i(A).$$
\indent It is known that there
is a unique non-empty compact subset $V$ of $\R^{N-1}$, called the
\textit{attractor} of the family ${\mathcal S}$, such that $L(V)=V$; see, Theorem 9.1 in Falconer \cite{Fa}. The set $V$ is called
the \textit{Sierpi\'nski gasket} in $\R^{N-1}$ of \textit{intrinsic boundary} $V_0:=\{p_1,\dots,p_N\}$.

Consider $H$ to be the convex hull of the set $V_0$ and observe that $S$ satisfies the
\textit{open set condition}, see \cite[p. 129]{Fa} taking $\Int\,(H)$ the interior of $H$, which is a non-empty
bounded open set such that
$$
\bigcup_{i=1}^{N}S_i(\Int\,(H))\subset \Int\,(H).
$$
\indent Since the above condition holds and $V$
is the attractor of ${\mathcal S}$, applying \cite[Theorem 9.3]{Fa}, we deduce that $V$ has
Hausdorff and box dimensions equal to the value of $d$ satisfying
\begin{equation}\label{for}
\sum_{i=1}^{N}\frac{1}{2^d}=1,
\end{equation}
and we also have ${\mathcal H}^d(V)\in (0,+\infty)$, where ${\mathcal H}^d$ is the $d$-dimensional Hausdorff
measure on $\R^{N-1}$. By relation \eqref{for}, we immediately get that $d=\log N/\log 2$. Let $\mu$ be the normalized restriction of the Hausdorff measure ${\mathcal
H}^d$ on
$\R^{N-1}$ to the subsets of $V$, so $\mu(V)=1$.\par
 We also recall, for completeness,
that if $0\leq d<\infty$, $0<\delta<\infty$ and $A\subset\RR^k$, then
$${\mathcal H}^d (A)=\lim_{\delta\rightarrow 0+}{\mathcal H}^d_\delta (A)=
\sup_{\delta > 0}{\mathcal H}^d_\delta (A),$$
where
$${\mathcal H}^d_\delta (A)=\frac{\pi^{d/2}}{\Gamma \left(\frac d2+1\right)}\inf\left\{ \sum_{j=1}^\infty \left(\frac{\mbox{diam}\, C_j}{2}\right)^d;\ A\subset\bigcup_{j=1}^\infty C_j,\ \mbox{diam}\, C_j\leq\delta\right\}$$
and $\Gamma (s)=\displaystyle\int_0^\infty e^{-x}x^{s-1}dx$ is Euler's Gamma function (see Evans and Gariepy \cite[p. 60]{evans} for details).


 Further, the following property
of $\mu$ will be useful in the sequel:
\begin{equation}\label{support}
\mu(B)>0,\hbox{ for every non-empty open subset $B$ of }V.
\end{equation}

In other words, the support of $\mu$ coincides with $V$; see, for instance, Breckner, R\u adulescu and Varga \cite{BRaduV} for more details.\par
\indent Denote by
$C(V)$ the space of real-valued continuous functions on $V$ and by
$$C_0(V):=\{u\in C(V)\mid u|_{V_0}=0\}.$$
The spaces $C(V)$ and $C_0(V)$ are endowed with the usual supremum
norm $\|\cdot\|_\infty$. For a function $u\colon V\to\R$ and for $m\in\N$
let
\begin{equation}\label{defWm}
W_m(u)=\left(\frac{N+2}{N}\right)^m\sum_{\underset{|x-y|=2^{-m}}{x,y\in
V_m}}(u(x)-u(y))^2,
\end{equation}
where $V_m:=L(V_{m-1})$, for $m\geq1$ and $V_*:=\bigcup_{m\geq 0} V_m$.\par
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 $\|\cdot\|_2$ norm. We now endow
$H_0^1(V)$ with the norm
$$\|u\|=\sqrt{W(u)}.$$

\indent In fact, there is an inner product defining this norm: for $u,v\in
H_0^1(V)$ and $m\in\N$ let
$${\mathcal W}_m(u,v)=\left(\frac{N+2}{N}\right)^m\sum_{\underset{|x-y|=2^{-m}}{x,y\in V_m}}(u(x)-u(y))(v(x)-v(y)).$$

\indent Set
$${\mathcal W}(u,v)=\lim_{m\to\infty} {\mathcal W}_m(u,v).$$

\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
\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}

\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 \indent 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
$$
\langle v,h\rangle=\int_Vv(x)h(x)d\mu,\qquad v\in L^2(V,\mu),\ h\in H^1_0(V).
$$
 Then $H^{-1}(V)$ is a Hilbert space. Then, 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 Finally, fix $\lambda>0$. Let $\alpha\colon
V\to\R$ and $f: \R\to\R$ be as in
Introduction. We say that a function $u\in H_0^1(V)$ is a
{\textit{weak solution} of} problem (\ref{N}) if
$${\mathcal W}(u,v)=-\lambda\int_V\alpha(x)f(u(x))v(x)d\mu,$$
for every $v\in H_0^1(V)$.\par

While we mainly work with the weak Laplacian, there is also a directly
defined version. We say that $\Delta_s$ is the \textit{standard Laplacian} of $u$ if $\Delta_su:V\to\R$ is continuous and
$$
\lim_{m\to\infty}\sup_{x\in V\setminus V_0}|(N+2)^m(H_mu)(x)-\Delta_su(x)|=0,
$$
\noindent where
$$
(H_mu)(x):=\sum_{\underset{|x-y|=2^{-m}}{y\in V_m}}(u(y)-u(x)),
$$
\noindent for $x\in V_m$. We say that $u\in C_0(V)$ is a {\it strong solution} of problem (\ref{N}) if $\Delta_su$ exists and is continuous for all $x\in V\setminus V_0,$ and
$$
\Delta_s u(x)=\lambda \alpha(x)f(u(x)),\,\,\,\,\,\forall\; x\in V\setminus V_0.
$$
\indent
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{
Since $f\colon\R\to\R$ is continuous and $\alpha\in C(V)$,
then  using 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}

Now, let $(X,\langle\cdot,\cdot\rangle)$ be a real Hilbert space and, for each $\gamma>0$, put
$$
B_\gamma:=\{u\in X:\|u\|^2\leq \gamma\}.
$$
Further, denote by $\Int\,(B_\gamma)$ the interior of $B_\gamma$.\par
The proof of our main result is obtained exploiting the following abstract theorem due to Ricceri \cite{ricceri} whose proof is entirely based on the results contained in \cite{ricceri2}.

\begin{theorem}\label{BMBlemma}
Let $J:X\rightarrow \RR$ be a sequentially weakly upper semicontinuous and G\^{a}teaux differentiable functional,
with $J(0_X)=0$. Assume that, for some $\gamma>0$, there exists a global maximum $\widehat{u}$ of $J|_{B_\gamma}$, such that
$$
\langle J'(\widehat{u}),\widehat{u}\rangle<2 J(\widehat{u}).
$$
Then, there exists an open interval $I\subseteq (0,+\infty)$ such that, for every $\lambda\in I$, the equation
$$
u=\lambda J'(u)
$$
has a non-zero solution lying in $\Int\,(B_\gamma)$.
\end{theorem}


\section{Proof of the Main Theorem}

Let us put $X:=H^{1}_0(V)$ endowed by the inner product
${\mathcal W}$ and define
$$
\widetilde{f}(t):=
\left\{\begin{array}{ll}
f(t) & \mbox{if}\,\,\,t\geq 0\\
0 & \mbox{if}\,\,\,t<0.
\end{array}\right.
$$
\noindent Since $f(0)=0$ it follows that $\widetilde{f}$ is a continuous function.\par
We consider the truncated problem
\begin{equation}\tag{$S_{\lambda, \alpha}^{\widetilde{f}}$}\label{N2}
\left\{
\begin{array}{l}
\Delta u(x)=\lambda \alpha(x)\widetilde{f}(u(x))\quad x \in V\setminus V_0, \\
u|_{V_0}=0,\\
\end{array}
\right.
\end{equation}
\noindent and set
$$
J(u):=-\int_V \alpha(x)\widetilde{F}(u(x))d\mu,
$$
 for every $u\in X$, where
$$\widetilde{F}(\xi):=\displaystyle\int_0^{\xi}\widetilde{f}(t)dt,$$
\noindent for every $\xi\in \RR$.\par
 By \cite[Proposition 4.5]{BRV} it follows that $J$ is a G\^{a}teaux differentiable and sequentially weakly  continuous functional with $J(0_X)=0$. Moreover, fixing $\lambda>0$, the weak solutions of \eqref{N2} are exactly the solutions $u\in X$ of the following equation
$$
u=\lambda J'(u)\,,
$$
see Proposition 2.19 in \cite{FaHu}.\par
 Further, Remark \ref{strong} ensures that every weak solution of problem (\ref{N2}) is a
strong one. Hence, exploiting the maximum principle proved by Strichartz in \cite[Theorem 2.1]{stripaper0}, every solution
$u\in X$ of \eqref{N2} is non-negative, so that $u$ also solves the original problem \eqref{N}.

\medskip
\noindent $(\rm i_1)\Rightarrow (\rm i_2)$


By hypothesis $({\rm{h}}_f)$, taking into account that $h_F$ is non-increasing in $(0,a]$ and since
$$
h_F'(\xi)=\displaystyle\frac{f(\xi)\xi-2F(\xi)}{\xi^3},\quad\forall\, \xi\in (0,a]
$$
we obtain
\begin{equation}\label{(1)}
f(\xi)\xi\leq 2F(\xi),
\end{equation}
for every $\xi\in (0,a]$.\par
 On the other hand, Fukushima and Shima proved in \cite{FuSc}
that the embedding
$$
(X,\|\cdot\|)\hookrightarrow (C_0(V),\|\cdot\|_\infty)
$$
is compact and
\begin{equation}\label{embeddingconstant}
\|u\|_\infty\leq(2N+3)\|u\|,
\end{equation}

\noindent for every $u\in X$.\par
Thus, let us fix
 $$r\in\left(0,\frac{a^2}{(2N+3)^2}\right],$$
 and denote $\gamma:=r^2.$\par
\indent By relations \eqref{(1)} and \eqref{embeddingconstant} it follows that
\begin{equation}\label{(2)}
\widetilde{f}(u(x))u(x)\leq 2\displaystyle \widetilde{F}(u(x)),
\end{equation}
for every $u\in B_\gamma$ and $x\in V$.\par
 Let $u_0\in X$ be a non-negative function in $V$ with $\|u_0\|=y>0$. Take $\varepsilon>y/\gamma$ and consider the function
$$
v_\varepsilon(x):=\frac{u_0(x)}{\varepsilon},\quad\forall\, x\in V.
$$
\indent Clearly $v_\varepsilon\in B_\gamma$ and
\begin{equation}\label{maj3}
J(v_\varepsilon)=-\int_{V_a^{\varepsilon}}\alpha(x)F(v_\varepsilon(x))d\mu> 0,
\end{equation}
where
$$
V_a^{\varepsilon}:=\{x\in V:0<v_\varepsilon(x)\leq a\},
$$
with $\mu(V_a^{\varepsilon})>0$.\par
\indent Now, let $\widehat{u}\in B_\gamma$ be a global maximum of $J$ in $B_\gamma$. Then, condition \eqref{maj3} ensures that $J(\widehat{u})>0$ and consequently
$$\displaystyle\max_{x\in V}\widehat{u}(x)> 0.$$
 \indent At this point, note that
\begin{equation}\label{(4)}
S_f:=\left\{x\in V:f(\widehat u(x))\widehat u(x)<2F(\widehat u(x))\right\}\neq \emptyset.
\end{equation}
\noindent Indeed, arguing by contradiction, if $S_f=\emptyset$, bearing in mind relation \eqref{(2)}, we would have
$$
f(\widehat u(x))\widehat u(x)=2F(\widehat u(x)),
$$
for every $x\in V$.\par
Then, since $h'_F(\xi)=0$ for every $$\xi \in A:=\left(0, \displaystyle\max_{x\in V}\widehat u(x)\right],$$ the function $h_F$ would be constant in $A$ against $(\rm i_1)$.\par
 Finally, since $\alpha$ is negative in $S_f$, relations \eqref{support} and \eqref{(4)} yield
\begin{equation}\label{Anto1}
-\int_{S_f}\alpha(x)f(\widehat u(x))\widehat u(x)d\mu<-2\int_{S_f} \alpha(x) F(\widehat u(x))d\mu.
\end{equation}

\indent Moreover
\begin{equation}\label{Anto2}
\int_{V\setminus S_f}\alpha(x)(f(\widehat u(x))\widehat u(x)-2F(\widehat u(x)))d\mu=0.
\end{equation}
\noindent Thus, by \eqref{Anto1} and \eqref{Anto2}, we write
$$
-\int_{V}\alpha(x)f(\widehat u(x))\widehat u(x)d\mu<-2\int_{V} \alpha(x) F(\widehat u(x))d\mu.
$$
\indent Bearing in mind that
$$
\langle J'(\widehat u), \widehat u\rangle=-\int_V\alpha(x)f(\widehat u(x))\widehat u(x)d\mu,
$$
the above inequality can be rewritten as
\begin{equation}\label{maieq}
\langle J'(\widehat u), \widehat u\rangle<2J(\widehat u).
\end{equation}
\indent On the other hand, if
$$
r\in\left(\frac{a}{(2N+3)},+\infty\right),
$$
by choosing $\gamma:={a^2}/{(2N+3)^2}$, and arguing as before, there exists a global maximum $\widehat{u}$ of $J$ in $B_\gamma$ such that condition \eqref{maieq} holds.\par
\indent Hence, Theorem \ref{BMBlemma} ensures that there exists an open interval $I\subseteq (0,+\infty)$ such that, for every $\lambda\in I$, problem \eqref{N} has a strong non-negative and non-zero solution, whose norm in $X$ is less than $\sqrt{\gamma}$. The conclusion follows.

\medskip
\noindent $(\rm i_1)\Leftarrow (\rm i_2)$


\indent Let us start recalling a preliminary fact on the spectrum of linear elliptic problems on the Sierpi\'nski gasket. More precisely, let $a:V\to \R$ be such that
\begin{itemize}
\item[{$({\rm{a}}_1)$}] \textit{$a(x)\geq 0$ in $V$ and $\displaystyle 0<\int_Va(x)d\mu<+\infty$},
\end{itemize} and consider the following elliptic eigenvalue problem
\begin{equation}\tag{$S_{\lambda, a}$}\label{Ngulp}
\left\{
\begin{array}{l}
\Delta u(x)+\lambda a(x)u(x)=0\quad x \in V\setminus V_0, \\
u|_{V_0}=0.\\
\end{array}
\right.
\end{equation}
\indent By \cite{Hu}, under the structural hypothesis $({\rm{a}}_1)$, it follows that problem (\ref{Ngulp}) possesses a sequence $\{\lambda_n\}$ of eigenvalues fulfilling
\begin{equation}\label{sidf}
0<\lambda_1<\lambda_2\leq \cdots\leq\lambda_n\leq\cdots,\,\,\,\,\,\,\,\lambda_n\rightarrow \infty\,\,\,{\rm as}\,\,\,\,n\rightarrow\infty,
\end{equation}
\noindent
see also \cite[pp. 563--564]{FaHu}.\par
From now on, we argue by contradiction. Hence, assume that there are two positive real constants $b$ and $c$ such that
$$
\frac{F(\xi)}{\xi^2}=c,\,\,\,\,\forall\, \xi\in (0,b].
$$
\indent Fixing
 $$r\in\left(0,\frac{b}{(2N+3)}\right], $$
 by $(\rm i_2)$, there exists an open interval $I$ such that, for every $\lambda\in I$, problem \eqref{N} admits a (strong) non-negative solution $u\in C_0(V)\setminus\{0_X\}$ such that
\begin{equation}\label{asd}
\|u\|<{r}\,.
\end{equation}
\indent In view of \eqref{embeddingconstant} and \eqref{asd}, we also obtain
$$
\|u\|_\infty< b,
$$
\noindent and it follows that
$$
f(u(x))=2cu(x),\,\,\,\,\forall\, x\in V.
$$
\indent Then, for every $\lambda\in I$, the linear problem
\begin{equation}\tag{$S_{\lambda, c\alpha}$}\label{Ngulp2}
\left\{
\begin{array}{l}
\Delta u(x)-2\lambda c\alpha(x)u(x)=0\quad x \in V\setminus V_0, \\
u|_{V_0}=0,\\
\end{array}
\right.
\end{equation}
admits a strong nonzero solution. This fact contradicts \eqref{sidf} and the proof is complete.\qed

\medskip
{\bf Acknowledgements.} The authors are grateful to the referee for the careful analysis of this paper and for constructive remarks.
 This paper was written when the first author was visiting professor at the University of Ljubljana in 2013. He expresses his gratitude to the host institution for warm hospitality.
  The manuscript was realized within the auspices of the GNAMPA Project 2013 titled {\it Problemi non-locali di tipo Laplaciano frazionario}.

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