Existence and Multiplicity of Solutions for Sublinear Schrödinger Equations with Coercive Potentials

Dong-Lun Wu

doi:10.1155/2019/9291534

Abstract

In this study, we consider the following sublinear Schrödinger equations−Δu+Vxu=fx,u,for x∈ℝN,ux⟶0,asu⟶∞,wherefx,usatisfies some sublinear growth conditions with respect touand is not required to be integrable with respect tox. Moreover,Vis assumed to be coercive to guarantee the compactness of the embedding from working space toLpℝNfor allp∈1,2∗. We show that the abovementioned problem admits at least one solution by using the linking theorem, and there are infinitely many solutions whenfx,uis odd inuby using the variant fountain theorem.

Highlights

Introduction and Main ResultsIn this study, we consider the following nonlinear Schrodinger equations:⎧⎨ − Δu + V(x)u f(x, u), for x ∈ RN, ⎩ u(x) ⟶ 0,(1) as |u| ⟶ ∞, where f(x, u) ∈ C(RN × R, R). e existence and multiplicity of solutions for (1) have attracted much attention of mathematicians
We show that the abovementioned problem admits at least one solution by using the linking theorem, and there are infinitely many solutions when f(x, u) is odd in u by using the variant fountain theorem
We show I possesses infinitely many critical points when F(x, t) is even in t, which are obtained by the variant fountain theorem

Summary

Introduction and Main Results

We consider the following nonlinear Schrodinger equations:. (1) as |u| ⟶ ∞, where f(x, u) ∈ C(RN × R, R). e existence and multiplicity of solutions for (1) have attracted much attention of mathematicians. Zhang et al [24] introduced some different sublinear conditions of f to obtain infinitely many nontrivial solutions for (1). In 2013, Bao and Han [5] considered (2), where g ≡ 0 and V is a sign-changing function while 0 lies in the spectrum of − Δ + V(x) and obtained infinitely many solutions for this resonant problem. Motivated by the abovementioned studies, we introduce some new sublinear growth conditions without weight functions to obtain the existence and multiplicity of solutions for (1). We recall the following coercive condition which is proposed by Wang and Han [14] to show the existence of infinitely many small solutions for a class of Kirchhoff equations with local sublinear nonlinearities by using the symmetric mountain pass theorem. En, F(x, t) satisfies the conditions of eorem 2, but the conditions in [1,2,3,4,5,6,7,8, 20, 23,24,25]

Preliminaries

Proof of Theorem 1