Multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical Sobolev exponents

Abstract

In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δ v= λu+ u p + εf( x), −Δ u= μv+ v q + δg( x) in Ω; u, v>0 in Ω; u= v=0 on ∂Ω, where Ω is a bounded domain in R N ( N⩾3); 0⩽ f, g∈L ∞(Ω); 1/( p+1)+1/( q+1)=( N−2)/ N, p, q>1; λ, μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ, μ∈(0, λ 1) and ε, δ∈(0, δ 0), where λ 1 is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ 0 is a positive number.