Polyharmonic systems involving critical nonlinearities with sign-changing weight functions

Abstract

This article concerns the existence of multiple solutions of the polyharmonic system involving critical nonlinearities with sign-changing weight functions (-Δ)mu = λƒ(x)|u|r-2u + β/β+γ h(x)|u|β-2 u|v|γ in Ω, (-Δ)mv = μg(x)|v|r-2v + γ/β+γ h(x)|u|β|v|γ-2v in Ω, Dku = Dkv = 0 for all |k| ≤ m - 1 on ∂Ω, where (-Δ)m denotes the polyharmonic operators, Ω is a bounded domain in ℝN with smooth boundary ∂Ω, m ∈ ℕ, N ≥ 2m + 1, 1 < r < 2 and β > 1, γ > 1 satisfying 2 < β + γ ≤ 2*m with 2*m = 2N/N-2m as a critical Sobolev exponent and λ, μ > 0. The functions ƒ, g and h : Ω → ℝ are sign-changing weight functions satisfying ƒ, g ∈ Lα(Ω) respectively. Using the variational methods and Nehari manifold, we prove that the system admits at least two nontrivial solutions with respect to parameter (λ, μ) ∈ ℝ2+ \ {(0, 0)}.