Existence and asymptotic behavior of solutions for Choquard equations with singular potential under Berestycki–Lions type conditions

Article type: Research Article

Authors: Zhu, Rui | Tang, Xianhua^{; *}

Affiliations: School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, P.R. China

Correspondence: [*] Corresponding author. E-mail: tangxh@mail.csu.edu.cn.

Abstract: We prove the existence and asymptotic behavior of solutions to the following problem: −Δu+V(x)u−g(x)u=(Iα∗F(u))f(u),x∈RN;u∈H1(RN), where g(x):=μ|x| is called the Coulomb potential, g(x):=β|x|2 is called the Hardy potential (the inverse-square potential). μ,β>0 are parameters, Iα:RN⟶R is the Riesz potential. Moreover, the nonlinearity f satisfies Berestycki–Lions type conditions which are introduced by Moroz and Van Schaftingen (Trans. Amer. Math. Soc. 367 (2015) 6557–6579). When μ∈(0,α(N−2)/2(α+1)) and β∈(0,α(N−2)2/4(2+α)), under some mild assumptions on V, we establish the existence and asymptotic behavior of solutions. Particularly, our results extend some relate ones in the literature.

Keywords: Choquard equation, ground state solution, Berestycki–Lions type conditions, Coulomb potential, Hardy potential

DOI: 10.3233/ASY-221798

Journal: Asymptotic Analysis, vol. 132, no. 3-4, pp. 427-450, 2023