High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness

Abstract

This article deals with the existence of high-energy positive solutions for the following coupled Schrödinger system with critical exponent: −Δu+V1(x)u=μ1u3+βuv2,x∈Ω,−Δv+V2(x)v=βu2v+μ2v3,x∈Ω,u,v∈D01,2(Ω)\left\{\begin{array}{l}-\Delta u+{V}_{1}\left(x)u={\mu }_{1}{u}^{3}+\beta u{v}^{2},\hspace{1em}x\in \Omega ,\\ -\Delta v+{V}_{2}\left(x)v=\beta {u}^{2}v+{\mu }_{2}{v}^{3},\hspace{1em}x\in \Omega ,\\ u,v\in {D}_{0}^{1,2}\left(\Omega )\end{array}\right. where Ω⊂R4\Omega \subset {{\mathbb{R}}}^{4} is an unbounded exterior domain, ∂Ω≠∅\partial \Omega \ne \varnothing , R4\Ω{{\mathbb{R}}}^{4}\backslash \Omega is bounded, V1,V2∈L2(Ω)∩Lloc∞(Ω){V}_{1},{V}_{2}\in {L}^{2}\left(\Omega )\cap {L}_{{\rm{loc}}}^{\infty }\left(\Omega ) are non-negative functions, and μ1,μ2{\mu }_{1},{\mu }_{2}, and β\beta are the positive constants. Combining variational methods and topological degree theory, we explore the existence of high-energy positive solutions to this system in the case of small perturbations of the source term, i.e., if ∣V1∣2+∣V2∣2{| {V}_{1}| }_{2}+{| {V}_{2}| }_{2} and R4\Ω{{\mathbb{R}}}^{4}\backslash \Omega are small enough in a prescribed sense. It is worth noting that our result still holds in the case Ω=R4\Omega ={{\mathbb{R}}}^{4}; hence, this article can be viewed as an extension of recent results for the Benci-Cerami problem in the framework of coupled Schrödinger systems with critical exponent in R4{{\mathbb{R}}}^{4}.

Keywords

• high-energy positive solutions
• coupled schrödinger system
• berestycki-lions condition
• brouwer degree
• critical exponent
• 35b33
• 35j47
• 35j57
• 35q55
• 47h11