Multiple positive solutions of strongly indefinite systems with critical Sobolev exponents and data that change sign

Abstract

In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems: −Δ v= λu+| u| p−1 u+ εf( x),−Δ u= μv+| v| q−1 v+ εg( x) in Ω; u>0, v>0 in Ω; u= v=0 on ∂Ω( ∗) , where Ω is a smooth bounded domain in R N ( N⩾3); f,g∈C 1( Ω ̄ ) ; p, q>1; λ, μ∈ R. For the subcritical and critical cases, we prove that problem ( ∗) has at least two positive solutions for any ε∈(0,ε ∗) and has no positive solutions for any ε>ε ∗(⩾ε ∗) . In the supercritical case, we find that the existence of solutions of problem ( ∗) for λ= μ=0 is closely related to the existence of nonnegative solutions of some linear elliptic system.