Abstract
In the present paper, we consider the following Hamiltonian elliptic system with Choquard’s nonlinear term −Δu+Vxu=∫ΩGvy/x−yβdygv in Ω,−Δv+Vxv=∫ΩFuy/x−yαdyfu in Ω,u=0,v=0 on ∂Ω,where Ω⊂ℝN is a bounded domain with a smooth boundary, 0<α<N, 0<β<N, and F is the primitive of f, similarly for G. By establishing a strongly indefinite variational setting, we prove that the above problem has a ground state solution.
Highlights
Introduction and Main ResultsIn this paper, we deal with the existence of ground state solutions for the following Hamiltonian elliptic system with Choquard’s nonlinear term:8 >>>>>>< >>>>>>: −Δu −Δv + + VðxÞu = VðxÞv = ð ΩGðvðyÞÞ jx − yjβ dygðvÞ inΩ, FðuðyÞÞ jx − yjα dy fðuÞ ð1Þ u = 0, v = 0 on ∂Ω, where Ω ⊂ RN is a bounded domain with a smooth boundary, N ≥ 3, 0 < α < N, 0 < β < N, and F is the primitive of f, for G
By establishing a strongly indefinite variational setting, we prove that the above problem has a ground state solution
We deal with the existence of ground state solutions for the following Hamiltonian elliptic system with Choquard’s nonlinear term: 8 >>>>>>< >>>>>>:
Summary
We deal with the existence of ground state solutions for the following Hamiltonian elliptic system with Choquard’s nonlinear term:. For N = 3, μ = 1, VðxÞ1, and f ðuÞ = u, the existence of ground states of (2) was obtained in [4] by variational methods. To the best of our knowledge, there is no work concerning the existence of ground state solutions to the Choquard-type Hamiltonian elliptic system. We are concerned with system (1) in whole space RN involving the Hamiltonian elliptic system with Choquard’s nonlinear term; it is a nonlocal problem, which brings about two obstacles. One is to check the linking structure, and the other one is to prove the boundedness of the corresponding (PS) sequences To avoid these obstacles, we shall deal with our problem in bounded domain.
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