Existence and localization of solutions for nonlocal fractional equations

TL;DR

This paper proves the existence of weak solutions for nonlocal fractional equations, including the fractional Laplacian, using variational methods and recent local minimum results in fractional Sobolev spaces.

Contribution

It establishes new existence theorems for fractional nonlocal equations with general integro-differential operators, extending previous results to broader classes of problems.

Findings

Existence of at least one weak solution for fractional equations.

Application of local minimum principles in fractional Sobolev spaces.

Development of variational framework for nonlocal fractional problems.

Abstract

This work is devoted to the study of the existence of at least one weak solution to nonlocal equations involving a general integro-differential operator of fractional type. As a special case, we derive an existence theorem for the fractional Laplacian, finding a nontrivial weak solution of the equation \begin{eqnarray*} \begin{cases} (-\Delta)^s u=h(x)f(u) & {\mbox{ in }} \Omega\\ u=0 & {\mbox{ in }} \mathbb{R}^n\setminus \Omega, \end{cases} \end{eqnarray*} where $h\in L^{\infty}_+(\Omega)\setminus\{0\}$ and $f:\mathbb{R}\rightarrow\mathbb{R}$ is a suitable continuous function. These problems have a variational structure and we find a nontrivial weak solution for them by exploiting a recent local minimum result for smooth functionals defined on a reflexive Banach space. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary.