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 \[ \begin{cases} (-\Delta)^s u = f(x,u) &\text{in }\Omega, \\u=0 &\text{in } {\mathbb R}^n \backslash \Omega. \end{cases} \] As far as we know, all these results are new.


35R09 Integro-partial differential equations
35R11 Fractional partial differential equations
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