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Mountain pass solutions for non-local elliptic operators. (English) Zbl 1234.35291

Summary: The purpose of this paper is to study the existence of solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. These equations have a variational structure and we find a non-trivial solution for them using the mountain pass theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. We prove this result for a general integrodifferential operator of fractional type and, as a particular case, we derive an existence theorem for the fractional Laplacian, finding non-trivial solutions of the equation

$\left\{\begin{array}{cc}{\left(-{\Delta }\right)}^{s}u=f\left(x,u\right)\hfill & \text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\hfill \\ u=0\hfill & \text{in}\phantom{\rule{4.pt}{0ex}}{ℝ}^{n}\setminus {\Omega }·\hfill \end{array}\right\$

As far as we know, all these results are new.

##### MSC:
 35R09 Integro-partial differential equations 35R11 Fractional partial differential equations