Raffaella Servadei, Enrico Valdinoci
Variational methods for non-local operators
of elliptic type
(381K, pdf)
ABSTRACT. In this paper we study the existence of non-trivial solutions for
equations driven by a non-local integrodifferential
operator~$\mathcal L_K$ with homogeneous Dirichlet boundary
conditions. More precisely, we consider the problem
$$ \left\{
egin{array}{ll}
\mathcal L_K u+\lambda u+f(x,u)=0 & {\mbox{ in }} \Omega\
u=0 & {\mbox{ in }} \RR^n\setminus \Omega\,,
\end{array}
ight.
$$
where $\lambda$ is a real parameter and the nonlinear term $f$
satisfies superlinear and subcritical growth conditions at zero and
at infinity. This equation has a variational nature, and so its
solutions can be found as critical points of the energy functional
$\mathcal J_\lambda$ associated to the problem. Here we get such
critical points using both the Mountain Pass Theorem and the Linking
Theorem, respectively when $\lambda<\lambda_1$ and $\lambda\geq
\lambda_1$\,, where $\lambda_1$ denotes the first eigenvalue of the
operator $-\mathcal L_K$.
As a particular case, we derive an existence theorem for the
following equation driven by the fractional Laplacian
$$ \left\{
egin{array}{ll}
(-\Delta)^s u-\lambda u=f(x,u) & {\mbox{ in }} \Omega\
u=0 & {\mbox{ in }} \RR^n\setminus \Omega\,.
\end{array}
ight.
$$
Thus, the results presented here may be seen as the extension
of some classical nonlinear analysis theorems to the case of fractional
operators.