Uniqueness of ground states for nonlinear Hartree equations

Abstract

We consider ground states of nonlinear Hartree equations with trapping potentials V(x), which can be described equivalently by positive minimizers of L2-critical Hartree energy functional. It is known that ground states exist if and only if the parameter a satisfies a<a*≔‖Q‖22, where Q is the unique ground state of ΔQ−Q+∫R4Q2(y)|x−y|2dyQ=0 in R4. In this paper, we prove the uniqueness of ground states as a ↗ a*, where the potential V(x) = p(x)h(x), 0<C≤p(x)≤1C, h(x) is homogeneous of degree q ≥ 2, and however H(y)=∫R4h(x+y)Q2(x)dx admits a unique and non-degenerate critical point.