Existence and localization of solutions for nonlocal fractional equations

Article type: Research Article

Authors: Molica Bisci, Giovanni^; | Repovš, Dušan

Affiliations: Dipartimento P.A.U., Università degli Studi Mediterranea di Reggio Calabria, Salita Melissari – Feo di Vito, 89124 Reggio Calabria, Italy. E-mail: gmolica@unirc.it | Faculty of Education, and Faculty of Mathematics and Physics, University of Ljubljana, POB 2964, Ljubljana, Slovenia 1001. E-mail: dusan.repovs@guest.arnes.si

Note: [] Corresponding author. E-mail: gmolica@unirc.it

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.

Keywords: fractional equations, multiple solutions, critical points results

DOI: 10.3233/ASY-141252

Journal: Asymptotic Analysis, vol. 90, no. 3-4, pp. 367-378, 2014