Existence and localization of solutions for nonlocal fractional equations

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.