Solutions to indefinite weakly coupled cooperative elliptic systems - Clapp, Mónica; Szulkin, Andrzej (Topological Methods in Nonlinear Analysis, 2021)
We study the elliptic system equation* cases -Δ u_1 - κ_1u_1 = μ_1|u_1|^p-2u_1 + λα|u_1|^α-2|u_2|^β u_1, -Δ u_2 - κ_2u_2 = μ_2|u_2|^p-2u_2 + λβ|u_1|^α|u_2|^β-2u_2, u_1,u_2D^1,2_0(Ω), cases equation* where Ω is a bounded domain in R^N, N≥ 3, κ_1,κ_2R, μ_1,μ_2,λ> 0, α,β> 1, and α + β = p2^*:=2N/(N-2). For p(2,2^*) we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for λ sufficiently large. If p=2^* and κ_1,κ_2> 0 we establish the existence of a ground state for λ sufficiently large if, either N5, or N=4 and neither κ_1 nor κ_2 are Dirichlet eigenvalues of -Δ in Ω.