Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type systems

Abstract

This paper deals with the following linearly coupled nonlinear Kirchhoff-type system:{−(a1+b1∫R3|∇u|2dx)Δu+μ1u=f(u)+βvin R3,−(a2+b2∫R3|∇v|2dx)Δv+μ2v=g(v)+βuin R3,u,v∈H1(R3), where ai>0,bi≥0,μi>0 are constants for i=1,2, β>0 is a parameter and f,g∈C(R,R). Under the general Berestycki–Lions type assumptions on f and g, we establish the existence of positive vector solutions and positive vector ground state solutions respectively by using variational methods. We also study the asymptotic behavior of these solutions as β→0+.