Existence and concentration of solutions for singularly perturbed doubly nonlocal elliptic equations

Abstract

We consider the existence and concentration of positive solutions to a singularly perturbed doubly nonlocal elliptic equation [Formula: see text] where [Formula: see text] is a parameter, [Formula: see text] are constants, [Formula: see text] and [Formula: see text] is an external potential, [Formula: see text] and [Formula: see text]. Under some suitable assumptions on [Formula: see text] and [Formula: see text], by using the penalization method, we prove that for [Formula: see text] small enough there exists a family of positive solutions which concentrate on the local minimum points of the potential [Formula: see text] as [Formula: see text].