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\begin{document}

\title*{A note on elliptic equations involving the critical Sobolev exponent}
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\author{Gabriele Bonanno, Giovanni Molica Bisci and Vicen\c{t}iu
R\u{a}dulescu}
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% your contribution title if the original one is too long
\institute{Gabriele Bonanno\at Department of Science for Engineering
and Architecture (Mathematics Section) Engineering Faculty,
University of Messina, 98166 - Messina, Italy,
\email{bonanno@unime.it} \and Giovanni Molica Bisci \at 
Dipartimento MECMAT, University of Reggio Calabria,
 Via Graziella, Feo di Vito, 89124 Reggio Calabria, Italy, \email{gmolica@unirc.it} \and Vicen\c{t}iu
R\u{a}dulescu \at Institute of Mathematics ``Simion Stoilow" of the
Romanian Academy, 014700 Bucharest, Romania. Department of
Mathematics, University of Craiova, 200585 Craiova, Romania,
\email{vicentiu.radulescu@imar.ro}}
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\abstract*{Each chapter should be preceded by an abstract (10--15 lines long) that summarizes the content. The abstract will appear \textit{online} at \url{www.SpringerLink.com} and be available with unrestricted access. This allows unregistered users to read the abstract as a teaser for the complete chapter. As a general rule the abstracts will not appear in the printed version of your book unless it is the style of your particular book or that of the series to which your book belongs.
Please use the 'starred' version of the new Springer \texttt{abstract} command for typesetting the text of the online abstracts (cf. source file of this chapter template \texttt{abstract}) and include them with the source files of your manuscript. Use the plain \texttt{abstract} command if the abstract is also to appear in the printed version of the book.}

\abstract{In this work we obtain some existence results for a class of
elliptic Dirichlet problems involving the critical Sobolev exponent and containing a parameter. Through a weak lower semicontinuity result and by using a critical point theorem for
differentiable functionals, the
existence of a precise open interval of positive eigenvalues for
which the treated problems admit at least one non-trivial weak solution is established. The attained results represent a more precise version of some contributions on the treated subject.}

\section{Introduction}
 In this paper we study the existence of one non-trivial weak solution for the following elliptic Dirichlet problem

 $$
({{P}_{\mu,\lambda}})\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{
\begin{array}{l}
-\Delta_p u= \mu \displaystyle\left(\int_{\Omega}|u(x)|^{p^*}dx\right)^{p/p^*-1}|u|^{p^*-2}u+\lambda f(x,u) \quad\rm in \quad \Omega
\\ u|_{\partial\Omega}=0,\\
\end{array}
\right.
$$

\noindent where $\Omega$ is a bounded domain in ${\erre}^N$ ($N\geq 2$) with smooth boundary $\partial \Omega$, $1<p<N$, $\Delta_pu:=\div(|\nabla u|^{p-2}\nabla u)$ stands for the usual $p$--Laplace operator, $p^{*}:=pN/(N-p)$ and $f:\Omega\times\erre\rightarrow\erre$ is a Carath\'{e}odory function satisfying the subcritical growth condition

  $$
|f(x,t)|\leq a_1+a_2|t|^{q-1},\,\,\,\,\,\forall\, (x,t)\in\Omega\times\erre,\eqno(\rm{h}_\infty)
$$
\noindent where $a_1,a_2$ are non-negative constants and $q\in ]1, pN/(N-p)[$. Finally, $\lambda$ and $\mu$ are two real parameters respectively positive and non-negative.\par
Problem $(P_{0,\lambda})$ has been extensively studied during the last few years, where the nonlinearity being a continuous
function provided with certain growth properties at zero and infinity, respectively. We just mention, in the large literature on the subject, the papers \cite{guo,hai,saint}; see also the recent monograph by
Krist\'aly, R\u adulescu and Varga \cite{KRV} as general reference
for this topic.\par
When $\mu\neq 0$, the classical variational approach cannot be applied due to the
presence of the term
$$
\displaystyle\left(\int_{\Omega}|u(x)|^{p^*}dx\right)^{p/p^*-1}|u|^{p^*-2}u.$$
Indeed, the classical Sobolev inequality, ensures that the
embedding of the space $W^{1,p}_0(\Omega)$ into the Lebesgue space $L^{p^*}(\Omega)$ is continuous but not compact. Due to this lack of compactness the classical methods cannot be used in order to prove the weak lower semicontinuity of the energy functional associated to $({{P}_{\mu,\lambda}})$. In our setting we overcome this difficulty through a lower semicontinuity result obtained by Montefusco in \cite{monte}. In this paper, bearing in mind the well-know inequality
\begin{equation}\label{P3}
\left(\int_\Omega|u(x)|^{p^*}dx\right)^{1/p^*}\leq \frac{1}{S^{1/p}}\left(\int_\Omega|\nabla u(x)|^{p}dx\right)^{1/p},\,\,\,\,\forall\, u\in W^{1,p}_0(\Omega)
\end{equation}
where $S$ is the best constant in the Sobolev inclusion $W^{1,p}_0(\Omega)\hookrightarrow L^{p^{*}}(\Omega)$ (see Section 2),
 fixing $\mu\in [0,S[$ and requiring a suitable behaviour of the nonlinearity $f$ at zero, we determine a precise open interval of positive parameters $\lambda$, for
which problem $({{P}_{\mu,\lambda}})$ admits at least one non-trivial weak solution in $W^{1,p}_0(\Omega)$.\par
The proof of our main results are based on a recent abstract critical point theorem proved by Bonanno and Candito in \cite[Theorem 3.1; part ({\rm a})]{BONCAN} which is substantially a refinement of the variational principle established by Ricceri in \cite{ricceri}; see also Bonanno and Molica Bisci \cite[Theorem 2.1; part ({\rm a})]{BMB}.\par

We explicitly observe that our results are a more precise form of the contributions obtained by Faraci and Livrea in \cite{faraci}. Indeed, in Theorem 3.1 of the cited paper, fixing $\mu\in [0,S[$, the authors proved the existence of a positive parameter $\nu^{*}_\mu$ such that, for every $\lambda\in ]0,\nu^{*}_\mu[$, the problem $({{P}_{\mu,\lambda}})$ admits at least one non-trivial weak solution. However, by using their approach, no concrete expression of this parameter is given.\par

Here, through a different strategy previously developed by Bonanno and Molica Bisci in \cite{BOMO}, an explicit value of the parameter $\nu^{*}_\mu$ is presented.
A particular case of our results (see Theorem \ref{Main1} and Remark \ref{Main2} below) reads as follows.
\begin{theorem}\label{Main3}
Let $f:\erre\to\erre$ be a continuous function satisfying the following subcritical growth condition
$$
|f(t)|\leq a_1+a_2|t|^{p-1},\,\,\,\,\,\forall\, t\in\erre,
$$
where $a_1,a_2$ are non-negative constants. Furthermore, assume that
$$
\lim_{\xi\rightarrow0^+} \frac{\di \int_0^\xi f(t)dt}{\xi^p}=+\infty.\eqno(\rm{h}_0')
$$
\noindent Then, for every
$\mu\in [0,S[$ there exists a positive number $\nu^{*}_\mu$ given by
$$
\nu^{*}_\mu:=\frac{S-\mu}{a_2S}\left(\frac{\omega_N}{\meas(\Omega)}\right)^{p/N},
$$
where $\omega_N$ is the volume of the unit ball in $\erre^N$, such that, for every $\lambda\in ]0,\nu^{*}_\mu[$,
the Dirichlet problem
$$
({{\widetilde{P}}_{\mu,\lambda}})\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{
\begin{array}{l}
-\Delta_p u= \mu \displaystyle\left(\int_{\Omega}|u(x)|^{p^*}dx\right)^{p/p^*-1}|u|^{p^*-2}u+\lambda f(u) \quad\rm in \quad \Omega
\\ u|_{\partial\Omega}=0,\\
\end{array}
\right.
$$
admits at least one non-trivial weak solution $u_{\lambda}\in W^{1,p}_0(\Omega)$. Moreover, one has that
$$
\lim_{\lambda\rightarrow 0^+}\|u_{\lambda}\|=0.
$$
\end{theorem}

\section{Abstract Framework and Main Results}
Let $\Omega$ be a bounded domain in ${\erre}^N$ with smooth
boundary $\partial \Omega$ and denote by $X$ the space
$W_{0}^{1,p}(\Omega)$ endowed with the norm
$$
\|u\|:= \Big(\int_{\Omega} |\nabla u(x)|^p dx\Big)^{1/p}.
$$
\noindent Fixing
$q\in[1,p^*[$, from the Sobolev embedding theorem, there
exists a positive constant $c_q$ such that
%
\begin{equation}\label{Poincare2}
\|u\|_{L^q(\Omega)}\leq c_q \|u\|\, ,\quad\, u\in X,
\end{equation}
%
\noindent and, in particular, the embedding $X\hookrightarrow L^q(\Omega)$ is compact. Moreover, the best constant that appears in inequality (\ref{P3}) is given by $S=1/c^p$, where
\begin{equation}\label{TA}
c=\frac{1}{N\sqrt{\pi}}\left(\frac{N!\Gamma\left(\di\frac{N}{2}\right)}{2\Gamma\left(\di\frac{N}{p}\right)\Gamma\left(\di N+1-\frac{N}{p}\right)}\right)^{1/N}\eta^{1-1/p},
\end{equation}
and
$$
\eta:=\displaystyle\frac{N(p-1)}{N-p},
$$
\noindent see, for instance, the quoted paper \cite{TA}. Let us define $\displaystyle F(x,\xi):= \int_{0}^{\xi} f(x,t) dt$, for every $(x,\xi)\in \Omega\times\erre$, and consider the
functional $E_{\mu,\lambda}: X \to \erre$
given by
$$
E_{\mu,\lambda}(u):= \Phi_{\mu}(u)-\lambda \Psi(u),\,\,\,\,u \in  X,
$$
where
$$
\Phi_{\mu}(u):=\frac{1}{p}\int_{\Omega}|\nabla u(x)|^p dx
-\frac{\mu}{p}\left(\int_\Omega|u(x)|^{p^*}dx\right)^{p/p^{*}},\,\,\,\,\,\,\,\Psi(u)
:= \int_{\Omega} F(x,u(x)) dx.
$$
\noindent Fixing $\mu\in [0,S[$, from the Sobolev inequality (\ref{P3}), it follows that
\begin{equation} \label{equivalence}
\left(\frac{S-\mu}{pS}\right)\|u\|^p\leq \Phi_\mu(u)\leq \frac{\|u\|^p}{p},
\end{equation}
\noindent for every $u\in X$. Furthermore, Montefusco, in \cite{monte}, proved that $\Phi_\mu$ is sequentially weakly lower semicontinuous for $\mu\in [0,S[$ and in this setting, since (\ref{P3}) holds, it is also a coercive functional.
Moreover, note that $E_{\mu,\lambda}\in C^{1}(X,\erre)$, and a critical point
$u\in X$ is a weak solution of the non-local problem
$$
-\Delta_p u= \mu \displaystyle\left(\int_{\Omega}|u(x)|^{p^*}dx\right)^{p/p^*-1}|u|^{p^*-2}u+\lambda f(x,u) \quad\rm in \quad \Omega.
$$
Our main tool in order to obtain the existence of one non-trivial solution to problem $({{P}_{\mu,\lambda}})$ is the following critical point theorem.

\begin{theorem}\label{BMB}
Let $X$ be a reflexive real Banach space, and let $\Phi,\Psi:X\to\R$
be two G\^{a}teaux differentiable functionals such that $\Phi$ is
(strongly) continuous, sequentially weakly lower semicontinuous and coercive. Further, assume that $\Psi$
is sequentially weakly upper semicontinuous. For every $r>\inf_X
\Phi$, put
$$
\varphi(r):=\inf_{u\in\Phi^{-1}(]-\infty,r[)}\frac{\displaystyle\left(\sup_{v\in\Phi^{-1}(]-\infty,r[)}\Psi(v)\right)-\Psi(u)}{r-\Phi(u)}.
$$
Then, for each $r>\inf_X\Phi$ and each
$\lambda\in\left]0,{1}/{\varphi(r)}\right[$, the restriction
of $J_\lambda:=\Phi-\lambda\Psi$ to
$\Phi^{-1}(]-\infty,r[)$ admits a global minimum, which is a
critical point $($local minimum$)$ of $J_\lambda$ in $X$.
\end{theorem}

\indent As pointed out in Introduction, this result is a refinement of the variational principle of Ricceri; see the quoted paper \cite{ricceri}.
Moreover, we recall that Theorem \ref{BMB} has been used in order to obtain some theoretical contributions on the existence of either three or infinitely many critical points for suitable  functionals defined on reflexive Banach spaces; see \cite{BONMAR} and \cite{BMB}. As consequences of the above cited results, on the vast literature on the subject, we mention here some recent works \cite{BMBR2d,BOMORa0,BOMORa1,BOMORa,BOMORa3} and \cite{DaMo,DaMo1,DaMo2} on the existence of weak solutions for some different classes of elliptic problems.\par
 \smallskip
The main result reads as follows.

\begin{theorem}\label{Main1}
Let $f:\Omega\times\erre\to\erre$ be a Carath\'{e}odory function with $f(x,0)\neq 0$ in $\Omega$, and satisfying condition $(\rm h_\infty)$. Then, for every
$\mu\in [0,S[$ there exists a positive number $\nu^{*}_\mu$ given by
$$
\nu^{*}_\mu:=q \sup_{\gamma>0}\left(\frac{{\gamma}^{p-1}}{\displaystyle qa_1c_1\left(\frac{pS}{S-\mu}\right)^{1/p}+a_2c_q^q\left(\frac{pS}{S-\mu}\right)^{q/p}{\gamma}^{q-1}}\right),
$$
such that, for every $\lambda\in ]0,\nu^{*}_\mu[$,
the following elliptic Dirichlet problem
$$
({{P}_{\mu,\lambda}})\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{
\begin{array}{l}
-\Delta_p u= \mu \displaystyle\left(\int_{\Omega}|u(x)|^{p^*}dx\right)^{p/p^*-1}|u|^{p^*-2}u+\lambda f(x,u) \quad\rm in \quad \Omega
\\ u|_{\partial\Omega}=0,\\
\end{array}
\right.
$$
 \noindent admits at least one non-trivial weak solution $u_{\lambda}\in X$. Moreover,
$$
\lim_{\lambda\rightarrow 0^+}\|u_{\lambda}\|=0
$$
and the function $\lambda\rightarrow E_{\lambda,\mu}(u_\lambda)$ is negative and decreasing in $]0,\nu^{*}_\mu[$.
\end{theorem}

\begin{proof}
Fix $\mu\in [0,S[$ and $\lambda\in ]0,\nu^{*}_\mu[$. Our aim is to apply
Theorem \ref{BMB} with $X=W^{1,p}_0(\Omega)$, $\Phi:=\Phi_\mu$ and $\Psi$ are the functionals introduced before. Since $\mu\in [0,S[$, $\Phi:X\to\R$ 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. Moreover, $\Phi$ is coercive and clearly $\displaystyle\inf_{u\in X}\Phi(u)=0$. Thanks to the growth condition $(\textrm{h}_\infty)$, one has that
\begin{equation}\label{inequality}
F(x,\xi)\leq a_1|\xi|+a_2\frac{|\xi|^{q}}{q},
\end{equation}
for every $(x,\xi)\in\Omega\times\erre$. Since $0<\lambda<\nu^{*}_\mu$,
there exists $\overline{\gamma}>0$ such that
\begin{equation}\label{n3}
\lambda<\nu^{\star}_\mu{(\overline{\gamma})}:=\frac{q\overline{\gamma}^{p-1}}{\displaystyle qa_1c_1\left(\frac{pS}{S-\mu}\right)^{1/p}+a_2c_q^q\left(\frac{pS}{S-\mu}\right)^{q/p}\overline{\gamma}^{q-1}}.
\end{equation}
\noindent Now, set $r\in ]0,+\infty[$ and consider the function
$$
\chi(r):=\frac{\displaystyle\sup_{u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}{r}.
$$
\noindent Taking into account (\ref{inequality}) it follows that
$$
\Psi(u)=\int_\Omega F(x,u(x))dx\leq a_1\|u\|_{L^1(\Omega)}+\frac{a_2}{q}\|u\|_{L^q(\Omega)}^q.
$$
Then, due to (\ref{equivalence}), we get
\begin{equation}\label{gio}
\|u\|< \left(\frac{pSr}{S-\mu}\right)^{1/p},
\end{equation}
\noindent for every $u\in X$ and $\Phi(u)<r$. Now, from (\ref{Poincare2}) and by using (\ref{gio}), for every $u\in X$ such that $\Phi(u)<r$ one has
$$
\Psi(u)< c_1a_1\left(\frac{pS}{S-\mu}\right)^{1/p}r^{1/p}+a_2\frac{c_q^q}{q}\left(\frac{pS}{S-\mu}\right)^{q/p}r^{q/p}.
$$
Hence
$$
\sup_{u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)\leq c_1a_1\left(\frac{pS}{S-\mu}\right)^{1/p}r^{1/p}+a_2\frac{c_q^q}{q}\left(\frac{pS}{S-\mu}\right)^{q/p}r^{q/p}.
$$
\noindent Then
\begin{equation}\label{n}
\chi(r)\leq c_1a_1\left(\frac{pS}{S-\mu}\right)^{1/p}r^{1/p-1}+a_2\frac{c_q^q}{q}\left(\frac{pS}{S-\mu}\right)^{q/p}r^{q/p-1},
\end{equation}
\noindent for every $r>0$.\par
\noindent Hence, in particular
\begin{equation}\label{n2}
\chi(\overline{\gamma}^p)\leq c_1a_1\left(\frac{pS}{S-\mu}\right)^{1/p}\overline{\gamma}^{1-p}+a_2\frac{c_q^q}{q}\left(\frac{pS}{S-\mu}\right)^{q/p}\overline{\gamma}^{q-p}.
\end{equation}
Now, observe that
$$
\varphi(\overline{\gamma}^p):=\inf_{u\in\Phi^{-1}(]-\infty,\overline{\gamma}^p[)}\frac{\displaystyle\left(\sup_{v\in\Phi^{-1}(]-\infty,\overline{\gamma}^p[)}\Psi(v)\right)-\Psi(u)}{r-\Phi(u)}\leq \chi(\overline{\gamma}^p),
$$
because $u_0\in\Phi^{-1}(]-\infty,\overline{\gamma}^p[)$ and $\Phi(u_0)=\Psi(u_0)=0$, where $u_0\in X$ is the identically zero function. In conclusion, bearing in mind (\ref{n3}), the above inequality together with (\ref{n2}), give
$$
\varphi(\overline{\gamma}^p)\leq \chi(\overline{\gamma}^p)\leq c_1a_1\left(\frac{pS}{S-\mu}\right)^{1/p}\overline{\gamma}^{1-p}+a_2\frac{c_q^q}{q}\left(\frac{pS}{S-\mu}\right)^{q/p}\overline{\gamma}^{q-p}<\frac{1}{\lambda}.
$$
In other words,
$$
\lambda\in \left]0,\frac{q\overline{\gamma}^{p-1}}{\displaystyle qa_1c_1\left(\frac{pS}{S-\mu}\right)^{1/p}+a_2c_q^q\left(\frac{pS}{S-\mu}\right)^{q/p}\overline{\gamma}^{q-1}}\right[\subseteq ]0,{1}/{\varphi(\overline{\gamma}^p)}[.
$$
Thanks to Theorem \ref{BMB}, there exists a function $u_\lambda\in\Phi^{-1}(]-\infty,\overline{\gamma}^p[)$ such that
$$
E'_{\mu,\lambda}(u_\lambda)=\Phi'(u_\lambda)-\lambda\Psi'(u_\lambda)=0,
$$
and, in particular, $u_\lambda$ is a global minimum of the restriction of $E_{\mu,\lambda}$ to $\Phi^{-1}(]-\infty,\overline{\gamma}^p[)$. Further, since $f(x,0)\neq 0$ in $\Omega$, the function $u_\lambda$ cannot be trivial, that is, $u_\lambda\neq 0$. Hence, for $\mu\in [0,S[$ and for every $\lambda\in ]0,\nu^{*}_\mu[$ the problem $({{P}_{\mu,\lambda}})$ admits a non-trivial solution $u_{\lambda}\in X$. From now, we argue in similar way of \cite[Theorem 3.1]{faraci} in order to prove that $\|u_{\lambda}\|\rightarrow 0$ as $\lambda\rightarrow 0^+$ and that
the function $\lambda\rightarrow E_{\mu,\lambda}(u_\lambda)$ is negative and decreasing in $]0,\nu^{*}_\mu[$.
The proof is complete.
\end{proof}
\begin{remark} \label{Main2}
{\rm
Also when $f(x,0)=0$ in $\Omega$ the statements of Theorem \ref{Main1} are still true if, in addition to assumption $(\rm h_\infty)$, the function $f$ satisfies the following hypothesis\par
\smallskip
{\it
\noindent There are a non-empty open set $D\subseteq \Omega$ and a set $B\subseteq D$ of positive Lebesgue measure such that
$$
\limsup_{\xi\rightarrow0^+} \frac{\inf_{x\in B}F(x,\xi)}{\xi^p}=+\infty
\,\,\,\,\,\,\,{\rm and}\,\,\,\,\,\,\liminf_{\xi\rightarrow0^+} \frac{\inf_{x\in D}F(x,\xi)}{\xi^p}>-\infty.\eqno(\rm{h}_0)
$$}
\noindent Condition ($\rm{h}_0$) ensures that the solution, achieved by using Theorem \ref{Main1}, is non-trivial.}
\end{remark}
\begin{remark} \label{Main0}
{\rm
In conclusion, we just mention that the technical approach adopted in this manuscript has been used in different settings in order to obtain existence and multiplicity results for several kinds of differential problems. For instance, by Bonanno, Molica Bisci and R\u{a}dulescu in \cite{BOMORa} for elliptic problems on compact Riemannian manifolds without boundary and by D'Agu\`{i} and Molica Bisci for an elliptic Neumann problem involving the $p$--Laplacian; see \cite{DaMo}.}
\end{remark}
\medskip
 {\bf Acknowledgements.}  V.~R\u adulescu acknowledges the support through  Grant CNCSIS PCCE--8/2010
{\it ``Sisteme diferen\c{t}iale \^{\i}n analiza neliniar\u{a} \c{s}i aplica\c{t}ii"}.
\input{referenc}
\end{document} 