Saddle point solutions for non-local elliptic operators

Alessio Fiscella

Abstract


The paper deals with equations driven by a non-local integrodifferential operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions.
These equations have a variational structure and we find a solution for them
using the Saddle Point Theorem. We prove this result for a general
integrodifferential operator of fractional type
and from this, as a particular case, one can derive an
existence theorem for the fractional Laplacian, finding
solutions of the equation
$$
\begin{cases}
(-\Delta)^s u=f(x,u) & {\mbox{in }} \Omega,\\
u=0 & {\mbox{in }} \mathbb{R}^n\setminus \Omega,
\end{cases}
$$
where the nonlinear term $f$ satisfies a linear growth condition.

Keywords


Integrodifferential operators; fractional Laplacian; variational techniques; Saddle Point Theorem; Palais-Smale condition

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