Semi-classical solutions for generalized quasilinear Schrödinger equations with nonlocal term and upper critical exponent

Abstract

In this paper, we focus on the following generalized quasilinear Schrödinger equation with nonlocal term and critical growth: $\begin{array}{rl}& -{ϵ}^2\mathrm{d}\mathrm{i}\mathrm{v}\left(g^2\right(u\left)\mathrm{\nabla }u\right)+{ϵ}^{2}g\left(u\right)g^{\mathrm{\prime }}\left(u\right)|\mathrm{\nabla }u{|}^2+V\left(x\right)u\\ & ={ϵ}^{\mu -N}[|x{|}^{-\mu }\ast |u{|}^{\alpha 2_{\mu }^{\ast }}]|u{|}^{\alpha 2_{\mu }^{\ast }-2}u+\lambda f\left(u\right),x\in {\mathbb{R}}^N,\end{array}$where $N\ge 3$, $ϵ>0$, $\lambda >0$, $2_{\mu }^{\ast }=\frac{2N-\mu }{N-2}$, $0<\mu <N$, $g:\mathbb{R}\to {\mathbb{R}}^{+}$ is a bounded $C^1$ even function, $g\left(0\right)=1$, $g^{\mathrm{\prime }}\left(s\right)\ge 0$ for all $s\ge 0$, $\underset{|s|\to +\mathrm{\infty }}{lim}\frac{g\left(s\right)}{|s{|}^{\alpha -1}}:=\text{β}>0$ for some $\alpha \in [1,2]$ and $(\alpha -1)g\left(s\right)\ge g^{\mathrm{\prime }}\left(s\right)s$ for all $s\ge 0$. By using the variational methods and Ljusternik–Schnirelmann category theory, we establish the existence and multiplicity of semi-classical solutions.

Keywords:

• Quasilinear Schrödinger equation
• nonlocal term
• variational method

2010 Mathematics Subject Classifications:

• 35J20
• 35J70
• 35P05
• 35P30

Acknowledgements

This work is supported in part by the National Natural Science Foundation of China (12026227; 12026228; 11801153; 12161033) and the Yunnan Province Applied Basic Research for Youths (2018FD085, 2018FD084) and the Yunnan Province Applied Basic Research for General Project (2019FB001, 2019FD091) and Technology Innovation Team of University in Yunnan Province and Youth Outstanding-notch Talent Support Program in Yunnan Province.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported in part by the National Natural Science Foundation of China (12026227; 12026228; 11801153; 12161033) and the Yunnan Province Applied Basic Research for Youths ((2018FD085, (2018FD084) and the Yunnan Province Applied Basic Research for General Project ((2019FB001, (2019FD091) and Technology Innovation Team of University in Yunnan Province and Youth Outstanding-notch Talent Support Program in Yunnan Province.