Positive ground state solutions for a class of Schrödinger–Poisson systems in R4 involving critical Sobolev exponent

Article type: Research Article

Authors: Khoutir, Sofiane^{a; b; *} | Chen, Haibo^a

Affiliations: [a] School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, P.R. China. E-mails: sofiane_math@csu.edu.cn, math_chb@csu.edu.cn | [b] Faculty of Mathematics, Laboratory AMNEDP, USTHB, Algiers 16111, Algeria

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

Abstract: In this paper, we study the following Schrödinger–Poisson system −Δu+u+λϕu=f(u)+u3,x∈R4,−Δϕ=u2,x∈R4, where λ>0 is a parameter, f∈C(R) and the nonlinear growth of u3 reaches the Sobolev critical exponent since 2∗=4 in dimension 4. Under some suitable assumptions on f, we establish the existence of a positive radial and non-radial ground state solutions for the above system by using variational methods. We also discuss the asymptotic behavior of solutions with respect to the parameter λ.

Keywords: Schrödinger–Poisson system, variational method, critical exponent, ground state solution

DOI: 10.3233/ASY-181471

Journal: Asymptotic Analysis, vol. 109, no. 1-2, pp. 91-109, 2018