Existence and Multiplicity of Solutions for a Coupled System of Kirchhoff Type Equations

Authors
• 1 Razi University, Kermanshah, Iran , Kermanshah (Iran)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer-Verlag
Publication Date
Oct 10, 2020
Volume
40
Issue
6
Pages
1831–1848
Identifiers
DOI: 10.1007/s10473-020-0614-7
Source
Springer Nature
Keywords
In this paper, we study the coupled system of Kirchhoff type equations {−(a+b∫ℝ3|∇u|2dx)Δu+u=2αα+β|u|α−2u|v|β,x∈ℝ3,−(a+b∫ℝ3|∇v|2dx)Δv+v=2βα+β|u|α|v|β−2v,x∈ℝ3,u,v∈H1(ℝ3),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\matrix{ { - \left( {a + b\int_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}{\rm{d}}x} } \right)\Delta u + u = {{2\alpha } \over {\alpha + \beta }}{{\left| u \right|}^{\alpha - 2}}u{{\left| v \right|}^\beta },\;x \in {\mathbb{R}^3},} \hfill \cr { - \left( {a + b\int_{{\mathbb{R}^3}} {{{\left| {\nabla v} \right|}^2}{\rm{d}}x} } \right)\Delta v + v = {{2\beta } \over {\alpha + \beta }}{{\left| u \right|}^\alpha }{{\left| v \right|}^{\beta - 2}}v,\;x \in {\mathbb{R}^3},} \hfill \cr {u,\;v \in {H^1}({\mathbb{R}^3}),} \hfill \cr } } \right.$$\end{document} where a,b > 0, α, β > 1 and 3 {vn<} α + β < 6. We prove the existence of a ground state solution for the above problem in which the nonlinearity is not 4-superlinear at infinity. Also, using a discreetness property of Palais-Smale sequences and the Krasnoselkii genus method, we obtain the existence of infinitely many geometrically distinct solutions in the case when α, β ≥ 2 and 4 < α + β < 6.