Semi-classical states for the Choquard equations with doubly critical exponents: Existence, multiplicity and concentration

Article type: Research Article

Authors: Su, Yu^a | Liu, Zhisu^{b; c; *}

Affiliations: [a] School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, Anhui 232001, PR China | [b] School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, PR China | [c] Center for Mathematical Sciences, China University of Geosciences, Wuhan, Hubei 430074, PR China

Correspondence: [*] Corresponding author. E-mail: liuzhisu@cug.edu.cn.

Abstract: In this paper, we are concerned with a class of Choquard equation with the lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality. We emphasize that nonlinearities with doubly critical exponents are totally different from the well-known Berestycki–Lions-type ones. Working in a variational setting, we prove the existence, multiplicity and concentration of positive solutions for such equations when the potential satisfies some suitable conditions. We show that the number of positive solutions depends on the profile of the potential and that each solution concentrates around its corresponding global minimum point of the potential in the semi-classical limit.

Keywords: Choquard equation, doubly critical exponents, semi-classical state, variational method, Moser iteration

DOI: 10.3233/ASY-221799

Journal: Asymptotic Analysis, vol. 132, no. 3-4, pp. 451-493, 2023