Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions

Abstract

We obtain uniqueness and nondegeneracy results for ground states of Choquard equations −∆u+u = |x| −1 * |u| p |u| p−2 u in R 3 , provided that p > 2 and p is sufficiently close to 2.

Concentration behavior of nonlinear Hartree-type equation with almost mass critical exponent

“…There have been a great deal of papers devote to equation (1.1) in different aspects. For results about the existence of positive ground states, one can see [1,13,14,18,19,20,26]; Cingolani [29] and Clap [31] investigated the existence of multiple solutions; for the uniqueness of the ground state, see for example [13,16,22,24,25]; and the semi-classical analysis results, see [28,30] and the references therein.…”

Existence and asymptotic behavior of the least energy solutions for fractional Choquard equations with potential well

“…Equation 4 with lower critical exponent p = N+ N also had been studied by Moroz and Van Schaftingen. 23 If N = 3 and = 2, Xiang 24 obtain the uniqueness and nondegeneracy results for the least energy solution of Equation 3 as p > 2 or p sufficiently close to 2. When V(x) is not a constant, positive solutions, sign-changing solutions, multibump solutions, multipeak solutions, normalize solutions, and so on are also studied for Equation 4, we refer the readers to Alves et al, Cingolani et al, Clapp and Salazar, Li and Ye, [25][26][27][28] and references therein.…”

High energy solutions of the Choquard equation

“…When Q(x) is a positive constant and 1 + α N ≤ p ≤ N +α N −2 , N ≥ 3, the existence of positive ground state solutions of (1.1) has been studied in many papers,see [8,10,11,12,13], for instance. In addition, uniqueness of positive solutions of the Choquard equations has also been widely discussed in recent years by a lot of papers [9,8,16,17,18]. In [17], T.Wang and Taishan Yi proved that the positive solution of (1.1) is uniquely determined, up to translation provided α = 2, p = 2, N = 3, 4, 5.…”