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%\title[Bifurcation analysis of a singular elliptic problem]
%{Bifurcation analysis of a singular elliptic problem modelling the equilibrium of anisotropic continuous media}
\title[Bifurcation analysis of a singular elliptic problem]
{Bifurcation analysis of a singular elliptic problem modelling the equilibrium of
anisotropic continuous media}

\author{Giovanni Molica Bisci and Vicen\c{t}iu D. R\u{a}dulescu}

\address[G. Molica Bisci]{Dipartimento Patrimonio, Architettura e Urbanistica, University of Reggio Calabria, Feo di Vito, 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{Critical point theory; variational methods; Ricceri's variational principle; singular elliptic equations; bifurcation points.}
\thanks{{\em 2010 Mathematics Subject Classification.} Primary: 35B32. Secondary: 35J60, 58E05.}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}%[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{Definition}{Definition}[section]
\newtheorem{example}{Example}[section]
\newtheorem{th}{Theorem}
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\newcommand{\erre}{{\mathbb R}}
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\def\essinf{\mathop {\rm ess\,inf}}
\def\esssup{\mathop {\rm ess\,sup}}
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\def\N{{\mathbb N}}
\def\di{\displaystyle}
\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{W^{1,p}(\Omega)}}
\def\xseg{\bar{x}}
\def\useg{\bar{u}}
\def\d{\mathop{\rm dist}}
\def\d{{\rm diam(\Omega)}}
\def\meas{\mathop{\rm meas}}
\begin{document}


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\begin{abstract}
In this work we obtain an existence result for a class of singular quasilinear
elliptic Dirichlet problems on a smooth bounded domain containing the origin. By using a Caffarelli-Kohn-Nirenberg
type inequality, a critical point result for
differentiable functionals is exploited, in order to prove the
existence of a precise open interval of positive eigenvalues for
which the treated problem admits at least one nontrivial weak solution. In the case of
 terms with a sublinear growth near the origin, we deduce the existence of solutions for small positive values of the parameter. Moreover, the corresponding solutions have smaller and smaller energies as the parameter goes to zero.
\end{abstract}
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\maketitle
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%INTRODUCTION%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
Stationary problems involving singular nonlinearities, as well as the associated evolution
equations, describe naturally several physical phenomena. At our best knowledge, the
first study in this direction is due to Fulks and Maybee \cite{FM}, who proved existence and
uniqueness results by using a fixed point argument; moreover, they showed that solutions
of the parabolic problem tend to the unique solution of the corresponding stationary equation.\par
 Further, nonlinear
singular boundary value problems arise in the context of chemical heterogeneous catalysts
and chemical catalyst kinetics, in the theory of heat conduction in electrically conducting
materials, singular minimal surfaces, as well as in the study of non-Newtonian fluids,
boundary layer phenomena for viscous fluids; we refer for more details to \cite{gr0,gr1,h1,h2}.

The problem we study in this paper corresponds to a model that describes several physical phenomena related to the equilibrium of anisotropic continuous media which possibly are somewhere ``perfect" insulators or ``perfect" conductors (see \cite[p.~79]{dautray}).

We are interested in the existence of solutions for the following singular elliptic Dirichlet problem
 \begin{equation}\left\{
\begin{gathered}
-\div(|x|^{-2a}\nabla u)= \lambda |x|^{-2b}f(u)\quad
\text{in }\Omega,\\
  u =0\quad \text{on }\partial \Omega, \tag{${D}_{\lambda}^{a,b}$}\label{P}
\end{gathered}\right.
\end{equation}
where $\lambda$ is a real positive parameter and $\Omega$ is a bounded domain in ${\erre}^N$ ($N>2$)
with smooth boundary $\partial \Omega$ containing the origin,
$a$ and $b$ are real numbers such that
$$
0\leq a<\frac{N-2}{2}\qquad{\rm and}\qquad a\leq b<a+1,
$$
 and $f:\erre\rightarrow\erre$ is a continuous function satisfying the following subcritical growth condition
\begin{equation}\tag{${\rm h}_{\infty}$}\label{S}
|f(t)|\leq a_1+a_2|t|^{r-1}\qquad\mbox{for all}\ t\in\erre,
\end{equation}
for some nonnegative constants $a_1,a_2$ and $r\in ]1, r_{\star}[$, where
$$
r_{\star}:=2\min\left\{\frac{(N-2b)}{N-2(a+1)},\frac{N}{N-2}\right\}.
$$
If
$r_\star=\displaystyle\frac{2N}{N-2}$
some concentration phenomena may occur, due to the
action of the non compact group of dilations in
$\erre^N$.

The most relevant feature in Problem \eqref{P} is due to the presence of the
singular coefficients $|x|^{-2a}$ and $|x|^{-2b}$,
which play a key role in the problem of existence
of  solutions when $0\in\overline\Omega$.

In 1984, Caffarelli, Kohn and Nirenberg proved in \cite{CKN84}, in the context of some more general inequalities, the following nice result:\par
\smallskip
\noindent \textit{Given $1<p<N$, for every $u\in C_0^{\infty}(\Omega)$, there exists a positive constant $C_{a,b}$ such that
\begin{equation}\left(\int_{\Omega}|x|^{-bq}|u(x)|^q\;dx\right)^{p/q}\leq C_{a,b}\int_{\Omega}|x|^{-ap}|\nabla u(x)|^p\;dx\,,\tag{$\rm CKN$}\label{CKNineq}\end{equation}
where
$$-\infty<a<\frac{N-p}{p},\;\;\; a\leq b\leq a+1,\;\;\; q=\frac{Np}{N-p(1+a-b)}\,,$$
and  $\Omega\subseteq\erre^N$ is an arbitrary open domain}.\par
\smallskip
 Note that the Caffarelli-Kohn-Nirenberg inequality \eqref{CKNineq} reduces to the classical Sobolev inequality (if $a=b=0$) and to the Hardy inequality (if $a=0$ and $b=1$).
Relation
\eqref{CKNineq} proves to be an important  tool in studying degenerate elliptic problems.
It is also related with the understanding of some important phenomena such as best constants, existence or nonexistence of extremal functions, symmetry properties of minimizers, compactness of minimizing sequences and concentration phenomena. We refer the reader to \cite{badiale, CM99,CM00,CW,gherad} and \cite{ghou,lieb,pucci,smets,WW} for some relevant applications of \eqref{CKNineq}; see also \cite{mikrad} for a Caffarelli-Kohn-Nirenberg type inequality with variable exponent and applications to PDE's.\par
In particular, in order to handle singular problems, various extensions of \eqref{CKNineq} were
achieved; see, for instance \cite{ACP}.
Moreover, in the last few years, existence, nonexistence and multiplicity of nonnegative solutions either for
\eqref{P} or some its variants, were obtained, where the nonlinear term is of pure power type; see \cite{AP,I,X}, and references
therein. In this direction, Ghergu and R\u adulescu \cite{gr} studied the following nonlinear singular elliptic equation
$$
-\div(|x|^{-2a}\nabla u)= K(x)|x|^{-bp}|u|^{p-2}u+\lambda g(x)\qquad {\rm in}\,\,\,\erre^N\,,
$$
where $g$ belongs to an appropriate weighted Sobolev space and $p$ denotes the Caffarelli-Kohn-Nirenberg critical exponent associated to the data $a,b$, and to the Euclidean dimension $N$. Under some natural assumptions
on the positive potential $K(x)$ the existence of some $\lambda_0>0$ was proved such that the above
problem has at least two distinct solutions provided that $\lambda \in\ (0,\lambda_0)$. The proof relies on
Ekeland's variational principle and on the mountain pass theorem without the Palais-Smale
condition, combined with a weighted variant of the celebrated Brezis-Lieb lemma (see \cite{brelie} for the original version of the Brezis-Lieb lemma).
 More recently, Krist\'{a}ly studied problem \eqref{P} proving the existence of infinitely many distinct nonnegative solutions
when $f$ is a continuous function oscillating either at the origin or at infinity; see \cite[Theorems 1, 2]{k}.

Motivated by this large interest on the subject in the current literature, in this paper we exploit a quoted critical point result for
differentiable functionals due to Ricceri (see Theorem \ref{BMB} below) in order to prove the
existence of a precise open interval of positive eigenvalues for
which the treated problem admits at least one nontrivial weak solution; see Theorem \ref{Main1} and Corollary \ref{Main3}. In addition, for $\lambda$  sufficiently small, the energy functional (namely $E_\lambda$) related to
the problem is negative and decreasing on the solutions; see Remark \ref{bifurcation}.


We also observe that when the nonlinear term is
sublinear at infinity, then the corresponding energy functional is
coercive. In this case, the existence of a solution (possibly trivial) comes from standard arguments; see Remark \ref{direct} for details. It is worth noticing that,
in our context, the potential may be also not coercive; see Example
\ref{Ex1} and Remark \ref{remfinale}.\par
We state here, as an example, the following
special case of our results; see also Remark \ref{rem2}.
\begin{theorem}\label{intro}
Let $0\leq b<1$ and $f:\R\to\R$ be a continuous function such that
$$
\lim_{t\rightarrow0^+} \frac{f(t)}{t}=+\infty
,\,\,\,\,\,\,{\rm and}\,\,\,\,\,\,\lim_{|t|\rightarrow
\infty}\frac{f(t)}{|t|^{s}}=0,\eqno(\rm{h}_0^{\infty})
$$
\noindent for some $$0\leq s<\displaystyle\frac{N+2(1-2b)}{N-2}.$$\par
 \noindent Then there exists
$\lambda^{\star}>0$ such that for all $\lambda\in
]0,\lambda^\star[$, the following singular Dirichlet problem
\begin{equation}\left\{
\begin{gathered}
-\Delta u= \lambda |x|^{-2b}f(u)\quad
{\rm in }\,\,\Omega,\\
  u =0\quad {\rm on }\,\,\partial \Omega, \tag{${D}_{\lambda}^{0,b}$}\label{EP}
\end{gathered}\right.
\end{equation}
admits at least one nontrivial weak solution $u_{\lambda}\in H^1_0(\Omega)$.\par
\noindent Moreover, we have $$\displaystyle\int_{\Omega} |\nabla u_{\lambda}(x)|^2 dx\rightarrow 0,$$ as
$\lambda\rightarrow 0^+$ and the mapping
$$
\lambda\mapsto \frac{1}{2}\Big(\int_\Omega |\nabla u_{\lambda}(x)|^2dx\Big)-\lambda\int_\Omega|x|^{-2b}\Big(\int_0^{u_{\lambda}(x)}f(t)dt\Big)dx,
$$
is negative and strictly decreasing in $]0,\lambda^{\star}[$.
\end{theorem}
The paper is organized as follows. In Section 2, we
recall some basic definitions and our main tool, while Section 3 is
devoted to our abstract results. A concrete example of application is pointed out.\par
 We refer to the monographs \cite{gr, KRV, mawwil} for several related topics and a careful analysis of our abstract framework.


\section{Preliminaries}

Let $\Omega$ be a bounded domain in ${\erre}^N$  with a smooth
boundary $\partial \Omega$ and containing the origin. Further, denote by ${{\mathcal{D}}}^{1,2}_a(\Omega)$ the completion of
$C_0^{\infty}(\Omega)$ with respect to the norm
$$
\|u\|_a= \Big(\int_{\Omega} |x|^{-2a}|\nabla u(x)|^2 dx\Big)^{1/2}.
$$
Then the embedding $j:{{\mathcal{D}}}^{1,2}_a(\Omega)\hookrightarrow L^r(\Omega;|x|^{-q})$ is continuous if
$$ 1\leq r\leq 2^{*}:=\frac{2N}{N-2}\qquad\mbox{and}\qquad q\leq (1+a)r+N\left(1-\frac{r}{2}\right),$$
where $L^r(\Omega;|x|^{-q})$ is the weighted Lebesgue space endowed with the norm
$$
\|u\|_{r,q}= \Big(\int_{\Omega} |x|^{-q}|u(x)|^r dx\Big)^{1/r}.
$$
We refer to \cite{CC,X} for more details.\par
 Hence, there exists a positive constant $K^{a}_{r,q}$ such that
\begin{equation}\label{equivalence}
\Big(\int_{\Omega} |x|^{-q}|u(x)|^r dx\Big)^{1/r}
\leq K^{a}_{r,q}\Big(\int_{\Omega} |x|^{-2a}|\nabla u(x)|^2 dx\Big)^{1/2},
\end{equation}
\noindent for every $u \in {{\mathcal{D}}}^{1,2}_a(\Omega).$\par
 Moreover, the embedding $j$ is compact if
\begin{itemize}
\item[$(\rm C)$] $\displaystyle 1\leq r< 2^{*}$\,\,\,{\rm and}\,\,\, $\displaystyle q< (1+a)r+N\left(1-\frac{r}{2}\right)$.
\end{itemize}

\indent Set $\displaystyle F(\xi):= \int_{0}^{\xi} f(t) dt$, for every $\xi\in\erre$. Consider the energy
functional $E_{\lambda}: {{\mathcal{D}}}^{1,2}_a(\Omega) \to \erre$
associated to \eqref{P} defined as follows
$$
E_{\lambda}(u)= \Phi(u)-\lambda \Psi(u)\qquad\mbox{for all} \ u \in {{\mathcal{D}}}^{1,2}_a(\Omega),
$$
where
$$
\Phi(u):=\frac{1}{2}\int_{\Omega} |x|^{-2a}|\nabla u(x)|^2 dx,\,\,\,\,\,\,\,\Psi(u)
:= \int_{\Omega} |x|^{-2b}F(u(x)) dx.
$$
 Then $\Phi$ is a G\^{a}teaux differentiable functional in ${{\mathcal{D}}}^{1,2}_a(\Omega)$ whose derivative is given by
$$
\Phi'(u)(v)=\int_\Omega |x|^{-2a}\nabla u(x)\cdot\nabla v(x) dx,
$$
\noindent for every $v\in {{\mathcal{D}}}^{1,2}_a(\Omega)$. Moreover, $\Phi$ is weakly lower semicontinuous and coercive.\par
\indent On the other hand, we have the following preliminary result.

\begin{lemma}\label{lemma}
Assume that $f$ verifies the growth condition \eqref{S}. Then the functional $\Psi:{{\mathcal{D}}}^{1,2}_a(\Omega)\rightarrow \erre$ is well defined, of class $C^1$, and sequentially weakly continuous.
\end{lemma}

{\it Proof}.
The functional $\Psi$ is well defined since, by using \eqref{equivalence} and exploiting $\rm (C)$, it follows that
$$
\Psi(u)\leq \left(a_1K_{1,2b}^{a}+\frac{a_2}{r}(K_{r,2b}^{a})^{r}\|u\|_{a}^{r-1}\right)\|u\|_{a}<+\infty,
$$
\noindent for every $u\in {{\mathcal{D}}}^{1,2}_a(\Omega)$.\par
\indent Further, standard arguments show that $\Psi$ is a continuously G\^{a}teaux differentiable functional whose G\^{a}teaux derivative is
$$
\Psi'(u)(v)=\int_\Omega |x|^{-2b}f(u(x))v(x)dx\qquad\mbox{for every $v\in {{\mathcal{D}}}^{1,2}_a(\Omega)$}.
$$

Now, we prove that $\Psi$ is sequentially weakly
continuous arguing by contradiction. Hence, let $\{u_k\}\subset {{\mathcal{D}}}^{1,2}_a(\Omega)$ be a sequence such that $u_k\rightharpoonup u$ in ${{\mathcal{D}}}^{1,2}_a(\Omega)$ but $\{\Psi(u_k)\}$ does not converges to $\Psi(u)$ as $n\rightarrow \infty$. Further, note that since
$$
r\in \left]1,\min\left\{\frac{2(N-2b)}{N-2(a+1)},2^*\right\}\right[,
$$
then $r$ and $2b$ verify relation $\rm (C)$. In other words
$$ 1< r< 2^{*}\qquad\mbox{and}\qquad
 2b< (1+a)r+N\left(1-\frac{r}{2}\right),$$
 see Remark \ref{VP} for details.
 Therefore, up to a subsequence, one can find a number $\varepsilon>0$ such that
$$
0<\varepsilon\leq |\Psi(u_k)-\Psi(u)|,
$$
for every $k\in\enne$ and $u_k\rightarrow u$ in $L^r(\Omega;|x|^{-2b})$.
\noindent Moreover, for every $k\in\enne$, we have $\vartheta_k\in ]0,1[$ such that
\begin{eqnarray*}\notag
      |\Psi(u_k)-\Psi(u)| &\leq & \int_\Omega |x|^{-2b}|F(u_k(x))-F(u(x))|dx \nonumber\\
                    &=& \int_\Omega |x|^{-2b}|f(u(x)+\vartheta_k(u_k(x)-u(x)))||u_k(x)-u(x)|dx .
\end{eqnarray*}
\noindent Thus, from the above inequality and \eqref{S}, it follows that
\begin{eqnarray*}\notag
      |\Psi(u_k)-\Psi(u)| &\leq & a_1\int_\Omega |x|^{-2b}|u_k(x)-u(x)|dx\nonumber\\
                    &+& a_2 \int_\Omega |x|^{-2b}|u(x)+\vartheta_k(u_k(x)-u(x))|^{r-1}|u_k(x)-u(x)|dx.
\end{eqnarray*}
Next, we employ the H\"{o}lder inequality (with weight $|x|^{-2b}$) and the following elementary property: {for every real number $s>0$, then there exists a constant $c_s>0$ such that
$$
(\alpha+\beta)^s\leq c_s(\alpha^s+\beta^s)\qquad\mbox{for all $\alpha,\beta\in (0,\infty)$.}
$$

\noindent Hence (note that $r>1$) we can write
\begin{eqnarray*}\notag
      \int_\Omega |x|^{-2b}|u(x)+\vartheta_k(u_k(x)-u(x))|^{r-1}|u_k(x)-u(x)|dx  &\leq & \nonumber\\
          \int_\Omega |x|^{-2b}(|u(x)|+|u_k(x)-u(x))|)^{r-1}|u_k(x)-u(x)|dx  &=& \nonumber\\
          c_{r-1}\left(\int_\Omega |x|^{-2b}|u(x)|^{r-1}|u_k(x)-u(x)|dx+\int_\Omega |x|^{-2b}|u_k(x)-u(x)|^{r}dx\right)  &\leq& \nonumber\\
          c_{r-1}\left(\int_\Omega |x|^{-2b}|u(x)|^{r}dx\right)^{{(r-1)}/{r}}\left(\int_\Omega |x|^{-2b}|u_k(x)-u(x)|^{r}dx\right)^{{1}/{r}} &+& \nonumber\\
          c_{r-1}\int_\Omega |x|^{-2b}|u_k(x)-u(x)|^{r}dx  &=& \nonumber\\
          c_{r-1}\left(\|u\|_{r,2b}^{r-1}+\|u_k-u\|_{r,2b}^{r-1}\right)\|u_k-u\|_{r,2b}.&& \nonumber
\end{eqnarray*}
Then
\begin{eqnarray*}\notag
      0<\varepsilon\leq |\Psi(u_k)-\Psi(u)| &\leq & a_1\|u_k-u\|_{1,2b}\nonumber\\
                    &+& c_{r-1}a_2 \left(\|u\|_{r,2b}^{r-1}+\|u_n-u\|_{r,2b}^{r-1}\right)\|u_k-u\|_{r,2b}.
\end{eqnarray*}
Since $\{u_k\}$ strongly converges to $u$ in $L^r(\Omega;|x|^{-2b})$ (so $u_k\rightarrow u$ in $L^1(\Omega;|x|^{-2b})$), the right-hand side tends to zero as $k\rightarrow \infty$, contradicting $\varepsilon>0$.
\qed

\smallskip
Fix $\lambda>0$. A function $u:\Omega\rightarrow \erre$ is said to be a \textit{weak
solution} of \eqref{P} if $u\in {{\mathcal{D}}}^{1,2}_a(\Omega)$ and
$$
\int_\Omega |x|^{-2a}\nabla u(x)\cdot\nabla v(x) dx=\lambda\int_\Omega |x|^{-2b}f(u(x))v(x)dx\qquad
\mbox{for all $v\in {{\mathcal{D}}}^{1,2}_a(\Omega)$}.
$$
Hence, from the above remarks, the critical points of $E_{\lambda}$
are exactly the weak solutions of \eqref{P}.\par
 Our main tool to prove the existence of a nontrivial weak solution for problem \eqref{P} is the following critical point theorem due to Ricceri \cite{ricceri}.

\begin{theorem}\label{BMB}
Let $X$ be a reflexive real Banach space and $\Phi,\Psi:X\to\R$
be two continuously G\^{a}teaux differentiable functionals such that $\Phi$ is
sequentially weakly lower semicontinuous and coercive and $\Psi$
is sequentially weakly upper semicontinuous. For every $\rho>\inf_X
\Phi$, set
$$
\varphi(\rho):=\inf_{u\in\Phi^{-1}(]-\infty,\rho[)}\frac{\displaystyle \sup_{v\in\Phi^{-1}(]-\infty,\rho[)}\Psi(v)-\Psi(u)}{\rho-\Phi(u)}\,.
$$
Then, for every $\rho>\inf_{u\in X}\Phi(u)$ and every
$\lambda\in\left]0,{1}/{\varphi(\rho)}\right[$, the restriction
of the functional $E_\lambda:=\Phi-\lambda\Psi$ to
$\Phi^{-1}(]-\infty,\rho[)$ admits a global minimum, which is a
critical point $($local minimum$)$ of $E_\lambda$ in $X$.
\end{theorem}
See also \cite[Theorem 2.1, part ({\rm a})]{BMBR2d}.

\section{Small energy solutions}

With the previous notations, the main abstract result of this paper reads as follows.

\begin{theorem}\label{Main1}
Let $f:\erre\to\erre$ be a continuous function satisfying condition $(\rm h_\infty)$.
Assume that
$$
\limsup_{\xi\rightarrow0^+} \frac{F(\xi)}{\xi^2}=+\infty,
\,\,\,\,\,\,\,{\rm and}\,\,\,\,\,\,\liminf_{\xi\rightarrow0^+} \frac{F(\xi)}{\xi^2}>-\infty.
$$

\noindent Then there exists a positive number $\lambda^{\star}$ given by
$$
\lambda^{\star}:=r \sup_{\gamma>0}\left(\dfrac{{\gamma}}{\displaystyle ra_1\sqrt{2}K_{1,2b}^{a}+a_22^{r/2}(K_{r,2b}^{a})^{r}{\gamma}^{r-1}}\right),
$$
such that for every $\lambda\in ]0,\lambda^{\star}[$,
the following elliptic Dirichlet problem\smallskip
 \begin{equation}\left\{
\begin{gathered}
-\div(|x|^{-2a}\nabla u)= \lambda |x|^{-2b}f(u)\quad
{\rm in }\,\,\Omega,\\
  u =0\quad {\rm on }\,\,\partial \Omega, \tag{${D}_{\lambda}^{a,b}$}\label{P}
\end{gathered}\right.
\end{equation}
 \noindent admits at least one nontrivial weak solution $u_{\lambda}\in {{\mathcal{D}}}^{1,2}_a(\Omega)$. Moreover,
$$
\lim_{\lambda\rightarrow 0^+}\|u_{\lambda}\|_a=0.
$$
and the function $\lambda\rightarrow E_{\lambda}(u_\lambda)$ is negative and strictly decreasing in $]0,\lambda^{\star}[$.
\end{theorem}

{\it Proof}.
Fix $\lambda\in ]0,\lambda^{\star}[$. Our aim is to apply
Theorem \ref{BMB} with $X:={{\mathcal{D}}}^{1,2}_a(\Omega)$ and where $\Phi$ and $\Psi$ are the functionals introduced in Section 2. Note that $X$ is a Hilbert space whose inner product $\langle\cdot,\cdot\rangle:X\times X\rightarrow \erre$ given by
$$
\langle u,v\rangle :=\int_{\Omega} |x|^{-2a}\nabla u(x)\cdot \nabla v(x)dx\qquad\mbox{for all}\  u,v\in X
$$
induces the norm $\|\cdot\|_a$.
Hence $X$ is a reflexive space. Now, we prove that problem $(P_\lambda)$ admits a weak solution $u_\lambda\in\Phi^{-1}(]-\infty,\bar\rho[)$, for a suitable $\bar\rho>0$.\par
 For our goal observe that $\Phi:X\to\R$ is a continuously G\^{a}teaux differentiable and sequentially weakly lower semicontinuous functional as well as the map $\Psi:X\to \erre$ is continuously G\^{a}teaux differentiable and sequentially weakly upper semicontinuous by Lemma \ref{lemma}.
  Moreover, $\Phi$ is coercive and $\inf_{u\in X}\Phi(u)=0$. Owing to $(\textrm{h}_\infty)$, if follows that
\begin{equation}\label{inequality}
F(\xi)\leq a_1|\xi|+a_2\frac{|\xi|^{r}}{r}\qquad\mbox{for all $\xi\in\erre$}.
\end{equation}

\noindent Since $0<\lambda<\lambda^{\star}$,
there exists $\bar{\gamma}>0$ such that
\begin{equation}\label{n3}
\lambda<\dfrac{r\bar{\gamma}}{\displaystyle ra_1\sqrt{2}K_{1,2b}^{a}+a_22^{r/2}(K_{r,2b}^{a})^{r}\bar{\gamma}^{r-1}}=:\lambda^{\star}_{\bar \gamma}.
\end{equation}
\noindent Now, set $\rho\in ]0,+\infty[$ and consider the function
$$
\chi(\rho):=\frac{\displaystyle\sup_{v\in\Phi^{-1}(]-\infty,\rho[)}\Psi(v)}{\rho}.
$$
\noindent Taking into account (\ref{inequality}) it follows that
$$
\Psi(v)
:= \int_{\Omega} |x|^{-2b}F(v(x)) dx\leq a_1\|v\|_{1,2b}+\frac{a_2}{r}\|v\|_{r,2b}^r.
$$
Then, for every $v\in X: \Phi(v)<\rho$, we get
\begin{equation}\label{gio}
\|v\|_a< \sqrt{2\rho},
\end{equation}
\noindent for every $v\in X$.\par
\noindent Now, from (\ref{equivalence}) and by using (\ref{gio}), for every $v\in X$ such that $\Phi(v)<\rho$, one has
$$
\Psi(v)< a_1K_{1,2b}^{a}\sqrt{2\rho}+\frac{a_22^{r/2}(K_{r,2b}^{a})^{r}}{r}\rho^{r/2}.
$$
Hence
$$
\sup_{v\in\Phi^{-1}(]-\infty,\rho[)}\Psi(v)\leq a_1K_{1,2b}^{a}\sqrt{2\rho}+\frac{a_22^{r/2}(K_{r,2b}^{a})^{r}}{r}\rho^{r/2}.
$$
\noindent Then
\begin{equation}\label{n}
\chi(\rho)\leq a_1K_{1,2b}^{a}\sqrt{\frac{2}{\rho}}+\frac{a_22^{r/2}(K_{r,2b}^{a})^{r}}{r}\rho^{r/2-1},
\end{equation}
\noindent for every $\rho>0$.\par
\noindent In particular, we deduce that
\begin{equation}\label{n2}
\chi(\bar{\gamma}^2)\leq \frac{a_1\sqrt{2}K_{1,2b}^{a}}{\bar \gamma}+\frac{a_22^{r/2}(K_{r,2b}^{a})^{r}}{r}\bar{\gamma}^{r-2}.
\end{equation}
At this point, observe that
$$
\varphi(\bar{\gamma}^2):=\inf_{u\in\Phi^{-1}(]-\infty,\bar{\gamma}^2[)}\frac{\displaystyle \sup_{v\in\Phi^{-1}(]-\infty,\bar{\gamma}^2[)}\Psi(v)-\Psi(u)}{\bar \gamma^2-\Phi(u)}\leq \chi(\bar{\gamma}^2),
$$
taking into account that $0_{X}\in \Phi^{-1}(]-\infty,\bar{\gamma}^2[)$ and $\Phi(0_{X})=\Psi(0_{X})=0$.\par
\noindent In conclusion, bearing in mind (\ref{n3}), the above inequality together with (\ref{n2}) yield
$$
\varphi(\bar{\gamma}^2)\leq \chi(\bar{\gamma}^2)\leq \frac{a_1\sqrt{2}K_{1,2b}^{a}}{\bar \gamma}+\frac{a_22^{r/2}(K_{r,2b}^{a})^{r}}{r}\bar{\gamma}^{r-2}< \frac{1}{\lambda}.
$$
In other words,
$$
\lambda\in \left]0,\dfrac{r\bar{\gamma}}{\displaystyle ra_1\sqrt{2}K_{1,2b}^{a}+a_22^{r/2}(K_{r,2b}^{a})^{r}\bar{\gamma}^{r-1}}\right[\subseteq ]0,{1}/{\varphi(\bar{\gamma}^2)}[.
$$
Thanks to Theorem \ref{BMB}, there exists a function $u_\lambda\in\Phi^{-1}(]-\infty,\bar{\gamma}^2[)$ such that
$$
E'_{\lambda}(u_\lambda)=\Phi'(u_\lambda)-\lambda\Psi'(u_\lambda)=0,
$$
and, in particular, $u_\lambda$ is a global minimum of the restriction of $E_{\lambda}$ to $\Phi^{-1}(]-\infty,\overline{\gamma}^2[)$.\par
\indent Next, we prove that the function $u_\lambda$ cannot be trivial ($u_\lambda\neq 0_X$). To this end, let us prove that
\begin{equation}\label{inf}
\limsup_{\|u\|\rightarrow 0}\frac{\Psi(u)}{\Phi(u)}=\limsup_{\|u\|\rightarrow 0}\frac{\displaystyle\int_{\Omega} |x|^{-2b}F(u(x)) dx}{\|u\|_a^2/2}=+\infty.
\end{equation}
 Thanks to our assumptions at zero, we can fix a sequence $\{\xi_k\}\subset \erre^{+}$ converging to zero and two constant $\sigma$, and $\kappa$ (with $\sigma>0$) such that
 $$
 \lim_{k\rightarrow \infty}\frac{\displaystyle\int_0^{\xi_k}f(t)dt}{\xi_k^2}=+\infty,
 $$
 and
 $$
 \int_0^\xi f(t)dt\geq \kappa \xi^2,
 $$
 \noindent for every $\xi\in [0,\sigma]$.\par
 \noindent Now, let us consider a closed ball $\bar B(x_0,\tau)\subset \Omega$ (not containing zero) for some $x_0\in \Omega$ and $\tau>0$ sufficiently small. Further, let $v\in C_0^{\infty}(\Omega)\subset X$ such that
 \begin{itemize}
 \item[i)] $v(x)\in [0,1]$, for every $x\in \Omega$;
 \item[ii)] $v(x)=1$, for every $x\in \bar B(x_0,\tau/2)$;
 \item[iii)] $v(x)=0$, for every $x\in \Omega\setminus B(x_0,\tau)$.
 \end{itemize}
 Finally,
fix $\varepsilon>0$ and consider a real positive number $\mu$ with

 $$
 \varepsilon<2\frac{\displaystyle \mu\int_{B(x_0,\tau/2)}|x|^{-2b}dx+\kappa\int_C|x|^{-2b}v(x)^2dx}{\|v\|^2_a},
 $$
 \noindent where, we set $C:=B(x_0,\tau)\setminus B(x_0,\tau/2)$. Note that the right-hand side of the above inequality is finite. Indeed since ${0}\notin B(x_0,\tau/2)$, then
 $$
 \int_{B(x_0,\tau/2)}|x|^{-2b}dx<+\infty.
 $$
 \noindent Moreover, since the injection $j:X\hookrightarrow L^2(\Omega;|x|^{-2b})$ is continuous, taking into account that
 $\displaystyle 1<2< 2^{*}$ and $\displaystyle 2b< 2(a+1)$,
 we deduce that there exists a  constant $K^{a}_{2,2b}>0$ (according to our notations) such that
$$
 \int_C|x|^{-2b}v(x)^2dx<\int_\Omega|x|^{-2b}v(x)^2dx\leq (K^{a}_{2,2b})^2\Big(\int_{\Omega} |x|^{-2a}|\nabla u(x)|^2 dx\Big)<+\infty.
 $$
\noindent Then, there is $\nu\in\enne$ such that $\xi_k<\delta$ and
$$
\int_0^{\xi_k}f(t)dt\geq \mu\xi_k^2,
$$
\noindent for every $k>\nu$.\par
\noindent At this point, for every $k>\nu$, and bearing in mind the properties of the function $v$, we obtain
\begin{eqnarray*}\notag
\frac{\Psi(\xi_k v)}{\Phi(\xi_k v)}      &=& 2\frac{\displaystyle \int_{B(x_0,\tau/2)}|x|^{-2b}\left(\displaystyle\int_0^{\xi_k}f(t)dt\right)dx+\int_C|x|^{-2b}F(\xi_kv(x))dx}{\|\xi_k v\|^2_a} \nonumber\\
                    &\geq& 2\frac{\displaystyle \mu\int_{B(x_0,\tau/2)}|x|^{-2b}dx+\kappa\int_C|x|^{-2b}v(x)^2dx}{\|v\|^2_a}>\varepsilon.
\end{eqnarray*}
Since $\varepsilon$ can be chosen arbitrarily large, it follows that
$$
\lim_{k\rightarrow \infty}\frac{\Psi(\xi_k v)}{\Phi(\xi_k v)}=+\infty,
$$
\noindent from which \eqref{inf} clearly follows. Hence, there exists a sequence $\{w_k\}\subset X$ strongly converging to zero, such that for every $k$ sufficiently large, $w_k\in \Phi^{-1}(]-\infty,\bar\gamma^2[)$, and
$$
E(w_k):=\Phi(w_k)-\lambda\Psi(w_k)<0.
$$
Since $u_\lambda$ is a global minimum of the restriction of $E_\lambda$ to $\Phi^{-1}(]-\infty,\bar \gamma^2[)$, we conclude that
$$
\Phi(u_\lambda)-\lambda\Psi(u_\lambda)<0=E_\lambda(0_X),
$$
hence $u_\lambda$ is not trivial.\par
\indent Now, we prove that
$\displaystyle
\lim_{\lambda\rightarrow 0^+}{\|u_{\lambda}\|_a}=0.$ Bearing in mind that $\Phi$ is a coercive functional and that $u_\lambda\in\Phi^{-1}(]-\infty,\bar \gamma^2[)$, for every $\lambda\in ]0,\lambda^{\star}_{\bar \gamma}[$, we obtain
$$
\|u_{\lambda}\|_a<\sqrt{2}\bar \gamma.
$$
\noindent Therefore, from $({\rm h}_{\infty})$, taking into account \eqref{equivalence} and the above inequality, it follows that there exists $M_{\bar\gamma}>0$ such that
\begin{equation}\label{bounded}
\left|\int_\Omega |x|^{-2b}f(u_{\lambda}(x))u_{\lambda}(x)dx\right|\leq M_{\bar\gamma},
\end{equation}
\noindent for every $\lambda\in ]0,\lambda^{\star}_{\bar\gamma}[$.\par
\noindent Now, $E'_\lambda(u_{\lambda})=0$, for every $\lambda\in ]0,\lambda^{\star}_{\bar\gamma}[$ and in particular $E'_\lambda(u_{\lambda})(u_{\lambda})=0$, that is,
$$
\|u_{\lambda}\|^2_a=\lambda\int_\Omega |x|^{-2b}f(u_{\lambda}(x))u_{\lambda}(x)dx,
$$
\noindent for every $\lambda\in ]0,\lambda^{\star}_{\bar\gamma}[$.\par
\noindent Then, from (\ref{bounded}), it follows that
$$
\lim_{\lambda\rightarrow 0^+}\|u_{\lambda}\|^2_a=\lim_{\lambda\rightarrow 0^+}\lambda\Psi'(u_{\lambda})(u_{\lambda})=0.
$$
\noindent Hence, as claimed,
$\displaystyle
\lim_{\lambda\rightarrow 0^+}{\|u_{\lambda}\|_a}=0.$\par

\indent Next, we observe that
$$
E_{\lambda}(u)=\lambda\left(\frac{\Phi(u)}{\lambda}-\Psi(u)\right),
$$
for every $u\in X$. Fix $0<\lambda_1<\lambda_2$ (with $\lambda_2<\lambda^{\star}_{\bar \gamma}$). Set
$$
m_{\lambda_1}:=\left(\frac{\Phi(u_{\lambda_1})}{\lambda_1}-\Psi(u_{\lambda_1})\right)=\inf_{u\in \Phi^{-1}(]0,\bar \gamma^2[)}\left(\frac{\Phi(u)}{\lambda_1}-\Psi(u)\right),
$$
and
$$
m_{\lambda_2}:=\left(\frac{\Phi(u_{\lambda_2})}{\lambda_2}-\Psi(u_{\lambda_2})\right)=\inf_{u\in \Phi^{-1}(]0,\bar \gamma^2[)}\left(\frac{\Phi(u)}{\lambda_2}-\Psi(u)\right).
$$
Clearly, as claimed before, $m_{\lambda_i}<0$ (for $i=1,2$), and $m_{\lambda_2}\leq m_{\lambda_1}$ thanks to $\lambda_1<\lambda_2$.
\noindent Then the mapping $\lambda\mapsto E_{\lambda}(u_{\lambda})$ is strictly decreasing in $]0,\lambda^{\star}[$ owing to
$$
E_{\lambda_2}(u_{\lambda_2})=\lambda_2m_{\lambda_2}\leq \lambda_2m_{\lambda_1}<\lambda_1m_{\lambda_1}=E_{\lambda_1}(u_{\lambda_1}).
$$ This concludes the proof.
\qed
\section{Additional properties and examples}
In this section we provide several related results and examples.

\begin{remark} \label{VP}
{\rm By simple direct computations, since
$$
0\leq a<\frac{N-2}{2},\,\,\,\,\,\,{\rm and}\,\,\,\,\,\, a\leq b<a+1,
$$
it follows that$$
2<\frac{2(N-2b)}{N-2(a+1)}.$$ Further, we observe that if, instead of $0\leq a<\displaystyle\frac{N-2}{2}$, we require the more restrictive condition $0\leq a<\displaystyle\frac{b}{N}(N-2)$, one has
$$
\frac{2(N-2b)}{N-2(a+1)}\in \left]2, 2^{*}\right[.
$$
Thus, in this special case, the above relation yields
$$
r_\star=\frac{2(N-2b)}{N-2(a+1)}.
$$
On the other hand if $a=b$, since
$$
\frac{N-2a}{N-2(a+1)}\geq\frac{N}{N-2},
$$
it follows that $r_\star=2^*$. Finally, we point out that the parameter $\lambda^{\star}$ in Theorem \ref{Main1} has the following expression
$$
\lambda^{\star}=\left\{
\begin{array}{l}
+\infty \quad\, {\rm if} \quad 1<r<2 \\
\displaystyle\frac{1}{a_2(K_{2,2b}^{a})^{2}} \quad {\rm if} \quad r=2 \\
\displaystyle \dfrac{{r\widetilde{\gamma}}_{\max}}{\displaystyle ra_1\sqrt{2}K_{1,2b}^{a}+a_22^{r/2}(K_{r,2b}^{a})^{r}\widetilde{\gamma}^{r-1}_{\max}} \quad {\rm if} \quad r\in \left]2,r_{\star}\right[, \\
\end{array}
\right.
$$
\noindent where
$$
{\widetilde{\gamma}}_{\max}:=\frac{1}{\sqrt{2}}\left(\frac{a_1rK_{1,2b}^{a}}{a_2(K_{r,2b}^{a})^{r}(r-2)}\right)^{1/(r-1)}.
$$
}
\end{remark}
\begin{remark} \label{direct}
{\rm From the above expressions, it follows that if $f$ is sublinear at infinity, Theorem \ref{Main1} ensures that
for all $\lambda>0$, the problem (\ref{P}) admits at least one nontrivial weak solution. We explicitly observe that, in this case, also the classical direct methods theorem (Tonelli's result) ensures the existence of at least one weak solution that anyway can be trivial.}
\end{remark}

\begin{remark} \label{bifurcation}
{\rm
 We observe that if $f(0)=0$, Theorem \ref{Main1} is a bifurcation result. Indeed, in this setting, it follows that the trivial solution solves problem \eqref{P} for every parameter $\lambda$. Hence, $\lambda=0$ is a bifurcation point for problem \eqref{P}, in the sense that the point $(0,0)$ belongs to the closure of the set
 $$
{\mathfrak{F}}:=\{(u,\lambda)\in {{\mathcal{D}}}^{1,2}_a(\Omega)\times]0,+\infty[: \mbox{ $u$ is a weak solution of \eqref{P}, $u\neq 0$}\}$$ in the space ${{\mathcal{D}}}^{1,2}_a(\Omega)\times\erre$. Otherwise, if $f(0)\neq 0$ evidently one has
 $$
 \lim_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^2}=+\infty.\eqno(\rm{h}_0')
 $$
 Hence, all the assumptions of Theorem \ref{Main1} are automatically verified and the conclusions follow.}
\end{remark}

\begin{remark} \label{rem2}
{\rm
Theorem \ref{intro} in Introduction easily follows from Theorem \ref{Main1} taking into account that the following $s$-sublinear assumption at infinity
$$\lim_{|t|\rightarrow
\infty}\frac{f(t)}{|t|^{s}}=0,$$
in which $s\in [0,s_\star)$, where $$s_\star:=\min\left\{\displaystyle\frac{N+2(a-2b+1)}{N-2(a+1)},\frac{N+2}{N-2}\right\},$$ implies the growth condition \eqref{S}. In addition, if
$$
\lim_{t\rightarrow0^+} \frac{f(t)}{t}=+\infty,
$$
also condition $(\rm{h}_0')$ holds true. Finally, bearing in mind Remark \ref{VP}, the parameter $\lambda^{\star}$ has the form
$$
\lambda^{\star}=\left\{
\begin{array}{l}
+\infty \quad\, {\rm if} \quad 0\leq s<1 \\
\displaystyle\frac{1}{\bar a_2(K_{2,2b}^{a})^{2}} \quad {\rm if} \quad s=1 \\
\displaystyle \frac{s-1}{2K_{1,2b}^{a}s}\left(\frac{\bar a_1(s+1)K_{1,2b}^{a}}{\bar a_2(K_{s+1,2b}^{a})^{s+1}(s-1)}\right)^{1/s} \quad {\rm if} \quad s\in \left]1,s_\star\right[, \\
\end{array}
\right.
$$
for suitable nonnegative real constants $\bar a_1,\bar a_2$.
}
\end{remark}


 The next result is an immediate consequence of Remark \ref{VP}.
\begin{corollary}\label{Main3}
Let $f:\erre\to\erre$ be a continuous function satisfying the following subcritical growth condition
\begin{equation}\tag{${\rm h}_{\infty}'$}\label{S23}
|f(t)|\leq a_1+a_2|t|^{r-1}\qquad\mbox{for all}\  t\in\erre,
\end{equation}
where $a_1,a_2$ are nonnegative constants and $r\in ]2, r_{\star}[$. Further, assume that condition $(\rm{h}_0')$ holds.
\noindent Then, for every
$$\lambda\in\left]0, \frac{r-2}{2K_{1,2b}^{a}(r-1)}\left(\frac{a_1rK_{1,2b}^{a}}{a_2(K_{r,2b}^{a})^{r}(r-2)}\right)^{\frac{1}{r-1}}\right[,$$
problem \eqref{P}
admits at least one nontrivial weak solution $u_{\lambda}\in {{\mathcal{D}}}^{1,2}_a(\Omega)$. Moreover,
$$
\lim_{\lambda\rightarrow 0^+}\|u_{\lambda}\|_a=0.
$$
and the function $\lambda\rightarrow E_{\lambda}(u_\lambda)$ is negative and strictly decreasing in $]0,\lambda^{\star}[$.
\end{corollary}


The next example deals with a nonlinearity $f$ that vanishes at zero. The existence of one nontrivial solution for the Dirichlet problem involving the map $f$ is achieved by using Corollary \ref{Main3}.
\begin{example} \label{Ex1}
{\rm  Let $\Omega$ be a bounded open subset of $\R^N$ (containing the origin) with $N\geq 3$, and consider the following problem
\begin{equation}\left\{
\begin{gathered}
-\div(|x|^{-2a}\nabla u)= \lambda |x|^{-2b}f(u)\quad
\text{in }\Omega,\\
  u =0\quad \text{on }\partial \Omega, \tag{${\widetilde{D}}_{\lambda}^{a,b}$}\label{SD2}
\end{gathered}\right.
\end{equation}
\noindent where $f(u):=|u|^{h-2}u+|u|^{s-2}u$, $1<h<2$ and $2<s<r_{\star}$.
Then, for every
$$\lambda\in\left]0, \frac{s-2}{2K_{1,2b}^{a}(s-1)}\left(\frac{sK_{1,2b}^{a}}{(K_{s,2b}^{a})^{s}(s-2)}\right)^{\frac{1}{s-1}}\right[,$$
\noindent problem (\ref{SD2}) admits at least one nontrivial weak solution $u_{\lambda}\in {{\mathcal{D}}}^{1,2}_a(\Omega)$ such that
$$
\lim_{\lambda\rightarrow 0^+}\|u_{\lambda}\|_a=0.
$$
To prove this, we can apply Corollary \ref{Main1} with
$$
f(t):=|t|^{h-2}t+|t|^{s-2}t,
$$
\noindent for every $t\in\R$. In fact, of course $f(0)=0$ and it is easy to verify that
$$
|f(t)|\leq 2(1+|t|^{s-1})\qquad\mbox{for all}\ t\in \erre.
$$
Then, condition $(\rm h_\infty')$ holds. Further, a direct computation shows that
$$
\lim_{\xi\rightarrow0^+} \frac{
F(\xi)}{\xi^2}\geq
\frac{1}{h}\left(\lim_{\xi\rightarrow0^+}
\frac{1}{\xi^{2-h}}\right)=+\infty.
$$
\noindent Hence, assumption $(\rm{h}_0')$ is verified and the conclusion follows.
}
\end{example}
\begin{remark} \label{remfinale}
{\rm
We point out that the energy functional $E_{\lambda}$ associated to problem $($\ref{SD2}$)$ is unbounded from below. Hence, since the functional $E_{\lambda}$ is not coercive, the classical direct method result cannot be applied to the case treated in Example \ref{Ex1}.
}
\end{remark}

\begin{remark} \label{remfinale2}
{\rm
In conclusion, we also note that a related bifurcation result (in respect to Theorem \ref{Main1}) for perturbed nonsingular elliptic Dirichlet problems and involving the classical Laplace operator $\Delta u:=\div(\nabla u)$ was successfully studied by Ricceri; see \cite[Theorem 1]{ricceri2} for completeness. On the contrary of the cited contribution, in Theorem \ref{Main1} we find a concrete expression of the parameter $\lambda^{\star}$ also in the laplacian case without singularities ($a=b=0$). More precisely, if $f$ is subcritical (hence $({\rm h}_{\infty})$ holds), under our assumptions at zero, problem $({D}_{\lambda}^{0,0})$ admits at least one nontrivial weak solution $u_{0,\lambda}\in H_0^{1}(\Omega)$ for every $\lambda\in ]0,\lambda^{\star}[$, in which
$$
\lambda^{\star}:=\left\{
\begin{array}{l}
+\infty \quad\, {\rm if} \quad 1<r<2 \\
\displaystyle\frac{1}{a_2c_2^2} \quad {\rm if} \quad r=2 \\
\displaystyle \frac{r-2}{2c_1(r-1)}\left(\frac{ra_1c_1}{a_2c_r^{r}(r-2)}\right)^{\frac{1}{r-1}} \quad {\rm if} \quad r\in \left]2,2^*\right[, \\
\end{array}
\right.
$$
\noindent where the numbers $c_r$ ($r\in [2,2^*[$) are the best Sobolev constants of the continuous inclusion $H^{1}_0(\Omega)\hookrightarrow L^r(\Omega)$.
Moreover, also in this case, we have $$\displaystyle\int_{\Omega} |\nabla u_{0,\lambda}(x)|^2 dx\rightarrow 0,$$ as
$\lambda\rightarrow 0^+$ and the mapping
$$
\lambda\mapsto \frac{1}{2}\Big(\int_\Omega |\nabla u_{0,\lambda}(x)|^2dx\Big)-\lambda\int_\Omega\Big(\int_0^{u_{0,\lambda}(x)}f(t)dt\Big)dx,
$$
is negative and strictly decreasing in $]0,\lambda^{\star}[$.}
\end{remark}

\medskip
\textbf{Acknowledgements.} The paper is realized with the auspices of the GNAMPA Project 2013: {\it Problemi non-locali di tipo Laplaciano frazionario}.
%V.~R\u adulescu acknowledges the support through Grant CNCS PCE-47/2011.

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\end{document}