Existence of Small-Energy Solutions to Nonlocal Schrodinger-Type Equations for Integrodifferential Operators in R-N

Abstract

We are concerned with the following elliptic equations: (-Delta)p,Ksu+V(x)

u

p-2u=lambda f(x,u), where (-Delta)p,Ks is the nonlocal integrodifferential equation with 0<s<1<p<+infinity, sp<N the potential function V:RN ->(0,infinity) is continuous, and f:RNxR -> R satisfies a Caratheodory condition. The present paper is devoted to the study of the L infinity-bound of solutions to the above problem by employing De Giorgi's iteration method and the localization method. Using this, we provide a sequence of infinitely many small-energy solutions whose L infinity-norms converge to zero. The main tools were the modified functional method and the dual version of the fountain theorem, which is a generalization of the symmetric mountain-pass theorem.