Existence of positive bound state solution for the nonlinear Schrödinger–Bopp–Podolsky system

Kaimin Teng,Yunxia Yan

doi:10.14232/ejqtde.2021.1.4

Abstract

In this paper, we study a class of Schrodinger–Bopp–Podolsky system. Under some suitable assumptions for the potentials, by developing some calculations of sharp energy estimates and using a topological argument involving the barycenter function, we establish the existence of positive bound state solution.

Highlights

In this paper, we consider the following Schrödinger–Bopp–Podolsky system−∆u + V(x)u + K(x)φu = Q(x)|u|p−1u in R3,−∆φ + ∆2φ = K(x)u2 in R3, (1.1)where p ∈ (3, 5), V(x), K(x) and Q(x) are positive functions such that lim V(x) = V∞ > 0, |x|→∞lim Q(x) = Q∞ > 0, lim K(x) = 0.This system appears when a Schrödinger field ψ = ψ(t, x) couple with its electromagnetic field in the Bopp–Podolsky electromagnetic theory
The purpose of this paper is to study the existence of bound state solution for system
Since ωρξ is a positive solution of problem (2.6), it follows that

Summary

Introduction

For any sign-changing critical point u of I∞, by standard argument, the following inequality holds true We are ready to consider the constrained minimization problem m := inf{I(u), u ∈ N }, we find that the relation between least energy m and m∞ holds and it is not achieved, we can not look for the ground state.

Objectives

Results

Conclusion