Existence and Multiplicity of Solutions to a Class of Fractional p-Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in ℝN

Yun-Ho Kim

doi:10.3390/math8101792

Abstract

We are concerned with the following elliptic equations: (−Δ)psv+V(x)|v|p−2v=λa(x)|v|r−2v+g(x,v)inRN, where (−Δ)ps is the fractional p-Laplacian operator with 0<s<1<r<p<+∞, sp<N, the potential function V:RN→(0,∞) is a continuous potential function, and g:RN×R→R satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a result on the existence of multiple solutions to the given problem by utilizing the well-known fountain theorem.

Highlights

The study of problems of elliptic type involving nonlocal fractional Laplacian or more general integro-differential operators has extensively been considered in light of the pure or applied mathematical theory to explain some concrete phenomena arising from the thin obstacle problem, crystal dislocation, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, soft thin films, phase transition phenomena, multiple scattering, image process, minimal surfaces and the Levy process [1,2,3,4,5,6], and the references therein
The considerable developments of the Bose-Einstein condensate activated the studies on the nonlinear waveforms of the nonlinear Schrödinger equations with external potentials and associated nonlinear partial differential equations
Motivated by huge interest in the current literature, exploiting variational methods, we investigate the existence of nontrivial weak solutions for the fractional p-Laplacian problems

Summary

Introduction

The study of problems of elliptic type involving nonlocal fractional Laplacian or more general integro-differential operators has extensively been considered in light of the pure or applied mathematical theory to explain some concrete phenomena arising from the thin obstacle problem, crystal dislocation, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, soft thin films, phase transition phenomena, multiple scattering, image process, minimal surfaces and the Levy process [1,2,3,4,5,6], and the references therein. The main aim of the present paper is to establish the existence of multiple solutions for Schrödinger-type problems in the case where the nonlinear term is concave-convex, by making use of the variational methods. Established the existence of at least two distinct nontrivial solutions for a Schrödinger-Kirchhoff type problem driven by the non-local fractional p(·)-Laplacian with the concave-convex nonlinearities when the convex term fulfilled the assumption (AR) and (Je), respectively. The first aim of the present article is to get the existence of two distinct nontrivial solutions for problem (1) for the case of a combined effect of concave-convex nonlinearities, provided that the condition on convex term g is weaker than (AR) and different from (Je), which is originally given in [48] even if the considered domain is bounded. Under appropriate conditions on g, we obtain several existence results of nontrivial weak solutions for problem (1) by utilizing the variational principle as the major tools

Preliminaries and Main Results

Conclusions