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Some Existence and Multiplicity Results For a Class of Quasilinear Elliptic p-Laplacian Systems

Volume 4, Number 2 (2013), 1 - 15

Some Existence and Multiplicity Results For a Class of Quasilinear Elliptic p-Laplacian Systems

Communicated By: 
Enrique Zuazua
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Abstract

We consider the existence and multiplicity of solutions for the following quasilinear elliptic system    −div(|∇u| p−2 ∇u)+m1(x)|u| p−2u=λFu(x,u,v)− 1 q Hu(x,u,v) x ∈ Ω, −div(|∇v| p−2 ∇v)+m2(x)|v| p−2v=λFv(x,u,v)− 1 q Hv(x,u,v) x ∈ Ω, with the boundary conditions |∇u| p−2 ∂u ∂n = μGu(x,u,v)+Ru(x,u,v) and |∇v| p−2 ∂v ∂n = μGv(x,u,v)+ Rv(x,u,v), where Ω ⊂ RN is a bounded smooth domain, p > q ≥ 2, λ, μ > 0 and the functions F, G, H, R, m1 and m2 satisfy some suitable conditions. Using the fibering maps and extracting Palais-Smale sequences in the Nehari manifold, we prove that the above system has at least two distinct positive solutions when the pair (λ, μ) belongs to a certain subset of R2.