Infinitely many solutions for a gauged nonlinear Schrödinger equation with a perturbation

Jiafa Xu,Jie Liu,Donal O'Regan

doi:10.15388/namc.2021.26.22496

Abstract

In this paper, we use the Fountain theorem under the Cerami condition to study the gauged nonlinear Schrödinger equation with a perturbation in R2. Under some appropriate conditions, we obtain the existence of infinitely many high energy solutions for the equation.

Highlights

We study the existence of infinitely many high energy solutions for the following gauged nonlinear Schrödinger equation with a perturbation in R2:
Motivated by the aforementioned works, in this paper, we study the existence of infinitely many high energy solutions under some appropriate conditions, which are weaker than the Ambrosetti–Rabinowitz conditions, and consider the effect of the parameters and the perturbation terms on the existence of solutions
(H5) implies that I is an even functional on E, and by Lemma 4 I satisfies all the conditions of Lemma 2

Summary

Introduction

We study the existence of infinitely many high energy solutions for the following gauged nonlinear Schrödinger equation with a perturbation in R2:. Problem (1) arises in the study of standing wave solutions for the gauged nonlinear Schrödinger equation iD0φ + (D1D1 + D2D2)φ + g(φ) = 0,. In [10], the authors studied the existence and multiplicity of the positive standing wave with f (u) + k(x), where the nonlinearity f behaves like exp(α|u|2) as |u| → ∞. They obtained a mountain-pass type solution when = 0. (AR) There exists μ > 6 such that 0 < μF (u) f (u)u for u ∈ R \ {0}

Preliminaries

Proof of Theorem 1