Multiplicity and concentration results for generalized quasilinear Schrödinger equations with nonlocal term

Abstract

This paper is concerned with the following generalized quasilinear Schrödinger equation with nonlocal term

$$\begin{aligned} -\varepsilon ^2\hbox {div}(g^2(u)\nabla u)\!+\!\varepsilon ^2g(u)g^\prime (u)|\nabla u|^2\!+\!V(x)u\!=\!\varepsilon ^{\mu -N}[|x|^{-\mu }\!*\!|u|^p]|u|^{p-2}u\!+\!f(u) \end{aligned}$$

for $x\in \mathbb {R}^{N}$, where $N\ge 3$, $\varepsilon >0$, $g: \mathbb {R}\rightarrow \mathbb {R}^+$ is a bounded $C^1$ even function, $g(0)=1$, $g^\prime (s)\ge 0$ for all $s\ge 0$, $\lim \nolimits _{|s|\rightarrow +\infty }\frac{g(s)}{|s|^{\alpha -1}}:=\beta >0$ for some $\alpha \in [1,2]$ and $(\alpha -1)g(s)\ge g^\prime (s)s$ for all $s\ge 0$, $2\alpha \le p< \frac{2\alpha (N-\mu )}{N-2}$, $0<\mu <N$. By using the variational methods and Ljusternik-Schnirelmann category theory, we establish the existence, multiplicity of semi-classical solutions and characterize the concentration behavior and exponential decay property for the above problem.