Alessio Fiscella, Raffaella Servadei, Enrico Valdinoci
A resonance problem for non-local elliptic operators
(54K, LaTeX)
ABSTRACT. In this paper we consider a resonance
problem driven by a non-local integrodifferential
operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions.
This problem has a variational structure and we find a solution for it
using the Saddle Point Theorem. We prove this result for a general
integrodifferential operator of fractional type
and from this, as a particular case, we derive an
existence theorem for the following fractional Laplacian equation
$$ \left\{
egin{array}{ll}
(-\Delta)^s u=\lambda a(x)u+f(x,u) & {\mbox{ in }} \Omega\
u=0 & {\mbox{ in }} \mathbb{R}^n\setminus \Omega\,,
\end{array}
ight.$$
when $\lambda$ is an eigenvalue of the related
non-homogenous linear problem
with homogeneous Dirichlet boundary data.