Infinitely many solutions for a class of sublinear fractional Schr\xf6dinger equations with indefinite potentials

Wen Guan,Xinan Hao,Da-Bin Wang

doi:10.1186/s13660-020-02326-8

Abstract

In this paper, we consider the following sublinear fractional Schrödinger equation: (−Δ)su+V(x)u=K(x)|u|p−1u,x∈RN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ (-\\Delta)^{s}u + V(x)u= K(x) \\vert u \\vert ^{p-1}u,\\quad x\\in \\mathbb{R}^{N}, $$\\end{document} where s, pin(0,1), N>2s, (-Delta)^{s} is a fractional Laplacian operator, and K, V both change sign in mathbb{R}^{N}. We prove that the problem has infinitely many solutions under appropriate assumptions on K, V. The tool used in this paper is the symmetric mountain pass theorem.

Highlights

Introduction and main resultIn this paper, we consider the following sublinear fractional Schrödinger equation:(– )su + V (x)u = K (x)|u|p–1u, x ∈ RN, (1.1)where s, p ∈ (0, 1), N > 2s, (– )s is a fractional Laplacian operator, K, V both change sign in RN and satisfy some conditions specified below.Problem (1.1) gives the following nonlinear field equation: ∂Ψ i = (–)sΨ + (1 + E)Ψ – K (x)|Ψ |p–1Ψ, x ∈ RN, t ∈ R+. (1.2) ∂tThe nonlinear field Eq (1.2) reflects the stable diffusion process of Lévy particles in random field
We prove that the problem has infinitely many solutions under appropriate assumptions on K, V
People found that this stable diffusion of Lévy process has a very important application in the mechanical system, flame propagation, chemical reactions in the liquid, and the anomalous diffusion of physics in the plasma

Summary

Introduction

1 Introduction and main result In this paper, we consider the following sublinear fractional Schrödinger equation: (– )su + V (x)u = K (x)|u|p–1u, x ∈ RN , (1.1) For fractional equations on the whole space RN , the main difficulty one may face is that the Sobolev embedding Hs(RN ) → Lq(RN ) is not compact for q ∈ [2, 2∗s ). Some authors [8, 10, 24, 31, 38, 50] considered fractional equations with the potential V satisfying the following conditions: (V ) V ∈ C(RN , R), infx∈RN V (x) ≥ V0 > 0 and, for each M > 0, meas{x ∈ RN : V (x) ≤ M} < ∞, where V0 is a constant and meas denotes Lebesgue measure in RN .

Results

Conclusion