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.