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

Giovanni Molica Bisci (Dipartimento P.A.U., Università degli Studi “Mediterranea” di Reggio Calabria, Italy)

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.

Full Text (PDF format)