Contents Online

# Mathematical Research Letters

## Volume 21 (2014)

### Number 2

### Sequences of weak solutions for fractional equations

Pages: 241 – 253

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n2.a3

#### Author

#### Abstract

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.