In this article, we consider the quasilinear Choquard equation with critical nonlinearity in ${\mathbb{R}}^N$: $-{\epsilon }^2\Delta u+V\left(x\right)u-{\epsilon }^{2}u\Delta u^2=\left({\int }_{{\mathbb{R}}^N}\frac{{\left|u\right|}^{22_{\mu }^{\ast }}}{{|x-y|}^{\mu }}dy\right){\left|u\right|}^{22_{\mu }^{\ast }-2}u+h(x,u)u,$where $0<\mu <N$, $N\ge 3$, $2_{\mu }^{\ast }=(2N-\mu )∕(N-2)$ is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality, $\epsilon$ is a real parameter. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Under suitable assumptions on $V$ and $h$, we investigate the existence and multiplicity of solutions for the above problem by using the mountain pass theorem and index theory. In order to overcome the lack of compactness, we apply the concentration-compactness principle.

在本文中，我们考虑了具有临界非线性的拟线性Choquard方程 ${\mathbb{[R}}^{ñ}$： $-{\epsilon }^2\Delta ü+V（X）ü-{\epsilon }^2ü\Delta {ü}^2=\left({\int }_{{\mathbb{[R}}^{ñ}}\frac{{\left|ü\right|}^{22_{\mu }^{\ast }}}{{|X-ÿ|}^{\mu }}dÿ\right){\left|ü\right|}^{22_{\mu }^{\ast }-2}ü+H（X，ü）ü，$哪里 $0<\mu <ñ$， $ñ\ge 3$， $2_{\mu }^{\ast }=（2ñ-\mu ）∕（ñ-2）$ 是Hardy–Littlewood–Sobolev不等式意义上的关键指数， $\epsilon$是一个实参。通过使用变量的变化，将拟线性方程简化为一个半线性方程，其相关函数在通常的Sobolev空间中得到了很好的定义，并满足了山口定理的几何条件。在适当的假设下$V$ 和 $H$，我们使用山路定理和指标理论研究上述问题的解的存在性和多重性。为了克服紧凑性的不足，我们应用了浓度-紧凑性原理。