### Saddle point solutions for non-local elliptic operators

#### 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.

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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