Uniqueness and non-degeneracy of ground states for Choquard equations with fractional Laplacian

Abstract

In this paper, we study the following Choquard equations with fractional Laplacian{(−Δ)su+u=(Iα⁎|u|p)|u|p−2uinRN,lim|x|→∞⁡u(x)=0,u∈Hs(RN), where (−Δ)s is the fractional Laplacian, Iα is the Riesz potential, s∈(0,1), 2s<N∈N, α∈(0,N) and p∈(N+αN,N+αN−2s). Via studying limiting profiles of ground states of the above problem, we establish the uniqueness and non-degeneracy of positive ground states as α is close to 0 and α is close to N respectively. As a by-product, some uniform regularity and decay estimates for the solutions to the fractional Choquard equation, which are also of interest and importance independently, are given by taking full advantage of the Bessel kernel and employing an iterative process.