$\mathbb{R}^2$中带临界指数增长的拟线性Choquard方程基态解的存在性和集中性

In this paper, we study the following perturbed quasilinear Choquard equation \begin{equation*} -\varepsilon^2\Delta u+V(x)u-\varepsilon^2\Delta (u^2)u=\varepsilon^{\mu-2}\big(\frac{1}{|x|^\mu}*F(u)\big)f(u),\quad x\in \ \mathbb{R}^2, \end{equation*} where $\varepsilon>0$ is a small parameter, $\frac{1}{|x|^\mu}$ with $0<\mu <2$ is the Riesz potential, $*$ is the convolution in $\mathbb{R}^2$, $V(x)\in C(\mathbb{R}^2, (0,+\infty))$, $F(u)$ is the primitive function of $f(u)$ and $f$ has critical exponential growth with respect to the Trudinger–Moser inequality. When $V$ verifies some assumptions, we apply variational methods and mountain pass theorem to obtain the existence and concentration behavior of positive ground state solutions for the above equation.