Existence and asymptotic behavior of standing wave solutions for a class of generalized quasilinear Schrödinger equations with critical Sobolev exponents

Article type: Research Article

Authors: Chen, Jianhua^a | Huang, Xianjiu^a | Qin, Dongdong^{b; *} | Cheng, Bitao^c

Affiliations: [a] Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, P.R. China. E-mails: cjh19881129@163.com, xjhuangxwen@163.com | [b] School of Mathematics and Statistics, Central South University, Changsha, Hunnan 410083, P.R. China. E-mail: qindd132@163.com | [c] School of Mathematics and Statistics, Qujing Normal University, Qujing, Yunnan 655011, P.R. China. E-mail: chengbitao2006@126.com

Correspondence: [*] Corresponding author. E-mail: qindd132@163.com.

Abstract: In this paper, we study the following generalized quasilinear Schrödinger equation −div(ε2g2(u)∇u)+ε2g(u)g′(u)|∇u|2+V(x)u=K(x)|u|p−2u+|u|22∗−2u,x∈RN, where N⩾3, ε>0, 4<p<22∗, g∈C1(R,R+), V∈C(RN)∩L∞(RN) has a positive global minimum, and K∈C(RN)∩L∞(RN) has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem with critical growth, and establish a phenomenon of exponential decay. Moreover, by Ljusternik–Schnirelmann theory, we also prove the existence of multiple solutions.

Keywords: Generalized quasilinear Schrödinger equation, existence, asymptotic behavior, ground state solutions, critical exponents

DOI: 10.3233/ASY-191586

Journal: Asymptotic Analysis, vol. 120, no. 3-4, pp. 199-248, 2020