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

(−Δ)^su=h(x)f(u) in Ω,

u=0 in Rⁿ\Ω, where h∈L^∞₊(Ω)\{0} and f :R→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.