A remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system

Abstract

In this work, motivated by [T.F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight function, Nonlinear. Anal. 68 (2008) 1733–1745], and using recent ideas from Brown and Wu [K.J. Brown, T.F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign changing weight function, J. Math. Anal. Appl. 337 (2008) 1326–1336], we prove the existence of nontrivial nonnegative solutions to the nonlinear elliptic system { − Δ p u + m ( x ) | u | p − 2 u = λ | u | γ − 2 u + α α + β c ( x ) | u | α − 2 u | v | β , x ∈ Ω , − Δ p v + m ( x ) | v | p − 2 v = μ | v | γ − 2 v + β α + β c ( x ) | u | α | v | β − 2 v , x ∈ Ω , u = 0 = v , x ∈ ∂ Ω . Here Δ p denotes the p -Laplacian operator defined by Δ p z = div ( | ∇ z | p − 2 ∇ z ) , p > 2 , Ω ⊂ R N is a bounded domain with smooth boundary, α > 1 , β > 1 , 2 < α + β < p < γ < p ∗ ( p ∗ = p N N − p if N > p , p ∗ = ∞ if N ≤ p ), ∂ ∂ n is the outer normal derivative, ( λ , μ ) ∈ R 2 ∖ { ( 0 , 0 ) } , the weight m ( x ) is a bounded function with ‖ m ‖ ∞ > 0 , and c ( x ) is a continuous function which changes sign in Ω ¯ .