Sequences of weak solutions for fractional equations

This work is devoted to study the existence of infinitely many weak solutions to nonocal equations involving a general integrodifferential operator of fractional type. These equations have a variational structure and we find a sequence of nontrivial weak solutions for them exploiting the $\mathbb{Z}_2$-symmetric version of the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. As a particular case, we derive an existence theorem for the fractional Laplacian, finding nontrivial solutions of the equation$$\begin{cases}(-\Delta)^s u = f(x,u) & {\mbox{ in }} \Omega, \\u=0 & {\mbox{ in }} \mathbb{R}^n \backslash \Omega\end{cases}$$As far as we know, all these results are new and represent a fractional version of classical theorems obtained working with Laplacian equations.

Keywords

nonlocal problems, fractional equations, mountain pass theorem

2010 Mathematics Subject Classification

Primary 35S15, 49J35. Secondary 45G05, 47G20.

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Published 1 August 2014