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.