Sequences of weak solutions for fractional equations

TL;DR

This paper proves the existence of infinitely many weak solutions for a class of nonlocal fractional equations using variational methods and symmetry arguments, extending classical results to fractional operators.

Contribution

It introduces new existence theorems for fractional equations with variational structure, including a fractional Laplacian case, using the Mountain Pass Theorem and fractional space analysis.

Findings

Existence of infinitely many weak solutions for fractional equations.

Application of the Mountain Pass Theorem in fractional settings.

New results for fractional Laplacian equations with nonlinearities.

Abstract

This work is devoted to study the existence of infinitely many weak solutions to nonlocal 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 $$ \left\{\begin{array}{ll} (-\Delta)^s u=f(x,u) & {\mbox{in}} \Omega\\ u=0 & {\mbox{in}} \erre^n\setminus \Omega. \end{array} \right. $$ As far as we know, all these results are new and represent a fractional version of classical theorems obtained working with Laplacian equations.