Existence and multiplicity of solutions for a class of (ϕ1, ϕ2)-Laplacian elliptic system in RN via genus theory. (July 2016)

• Record Type: Journal Article
• Authors: Wang, Liben; Zhang, Xingyong; Fang, Hui
• Is Part Of: Computers & mathematics with applications. Volume 72:issue 1(2016)
• Page Start: 110
• Page End: 130
• Publication Date: 2016-07
• Subjects: (ϕ1, ϕ2)-Laplacian system -- Genus theory -- Weak solution -- Critical point; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2016.04.034
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: In this paper, we investigate the following nonlinear and non-homogeneous elliptic system involving ( ϕ 1, ϕ 2 ) -Laplacian { − div ( ϕ 1 ( | ∇ u | ) ∇ u ) + V 1 ( x ) ϕ 1 ( | u | ) u = F u ( x, u, v ) in R N, − div ( ϕ 2 ( | ∇ v | ) ∇ v ) + V 2 ( x ) ϕ 2 ( | v | ) v = F v ( x, u, v ) in R N, ( u, v ) ∈ W 1, Φ 1 ( R N ) × W 1, Φ 2 ( R N ) with N ≥ 2, where the functions V i ( x ) ( i = 1, 2 ) are bounded and positive in R N, the functions ϕ i ( t ) t ( i = 1, 2 ) are increasing homeomorphisms from R + onto R +, and the function F is of class C 1 ( R N + 2, R ) and has a sub-linear Orlicz–Sobolev growth. By using the least action principle, we obtain that system has at least one nontrivial solution. When F satisfies an additional symmetric condition, by using the genus theory, we obtain that system has infinitely many solutions.