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Multiple positive solutions of strongly indefinite systems with critical Sobolev exponents and data that change sign. (English) Zbl 1163.35376
Summary: We study the existence of multiple positive solutions to some Hamiltonian elliptic systems: (*) -Δv=λu+|u| p-1 u+εf(x), -Δu=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 N (N3); f,gC 1 (Ω ¯); p,q>1; λ,μ. 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.
MSC:
35J60Nonlinear elliptic equations
35B33Critical exponents (PDE)