Ground State Solution for an Autonomous Nonlinear Schrödinger System

Min Liu,Jiu Liu,Andrea Scapellato

doi:10.1155/2021/1003941

Abstract

In this paper, we study the following autonomous nonlinear Schrödinger system (discussed in the paper), where λ , μ , and ν are positive parameters; 2 ∗ = 2 N / N − 2 is the critical Sobolev exponent; and f satisfies general subcritical growth conditions. With the help of the Pohožaev manifold, a ground state solution is obtained.

Highlights

Introduction and Main ResultIn this paper, we consider the following autonomous nonlinear Schrödinger system:8 >>< −Δu + μu = μf ðuÞ + λv, x ∈ RN, >>: −Δv + νv = jvj2∗ u, v ∈ H ÀRN Á, −2 v +
We study the following autonomous nonlinear Schrödinger system, where λ, μ, and ν are positive parameters; 2∗ = 2N/ðN − 2Þ is the critical Sobolev exponent; and f satisfies general subcritical growth conditions
Inspired by the above literatures, especially [6], we investigate the existence of ground state solution of system (1)

Summary

Introduction and Main Result

We consider the following autonomous nonlinear Schrödinger system:. >>:. V ∈ D1,2ðR4Þ, μ = V1ðxÞ, and ν = V2 ðxÞ satisfy the integral conditions and jujp−1u, λv, jvjq−1v and λu are replaced by μ1u3, βuv, μ2v3, and βu2v, respectively, Liu and Liu [4] proved that (2) has a positive solution. N ≥ 3, 1 < Chen and p< Zou [6] proved that (2) has a positive ground state solution under λ, μ, ν which satisfied certain conditions. By [9], we know that if f satisfies (f1)-(f4); equation (3) has a ground state solution. There are some recent studies on the ground state solutions for other types of Schrödinger equations or systems, for example, [6, 11]. In the bounded domain, the existence and the regularity of solutions to differential problems have been widely investigated by using tools of harmonic and real analysis and variational methods, for example, [12,13,14]

Preliminaries

Proof of Theorem 1

Findings

TN ðRN μT