We are concerned with a class of singularly perturbed quasilinear Schrodinger equations of the following form: \[ - {\varepsilon ^2}\Delta u - {\varepsilon ^2}\Delta ({u^2})u + V(x)u = h(u),{\text{ }}u > 0{\text{ in }}{\mathbb{R}^N}, \] where $\varepsilon $ is a small positive parameter, $N \ge 3$ and the nonlinearity $h$ is of critical growth. We construct a family of positive solutions ${u_\varepsilon } \in {H^1}({\mathbb{R}^N})$ of the above problem which concentrates around local minima of $V$ as $\varepsilon \to 0$ under certain assumptions on $h$. Our result especially solves the above problem in the case where $h(u) \sim \lambda {u^{q - 1}} + {u^{2 \cdot {2^ * } - 1}}{\text{ }}(2 0)$ and completes the study made in some recent works in the sense that, in those papers only the case where $h(u) \sim \lambda {u^{q - 1}} + {u^{2 \cdot {2^ * } - 1}}{\text{ }}(4 0)$ was considered. Moreover, our main results extend also the arguments used in Byeon and Jeanjean [14], which deal with Schrodinger equations with subcritical nonlinearities, to the quasilinear Schrodinger equations with critical nonlinearities.

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一类具有一般非线性的奇异摄动拟线性薛定谔方程的集中孤立波

我们关注一类奇异摄动拟线性薛定谔方程，其形式如下： \[ - {\varepsilon ^2}\Delta u - {\varepsilon ^2}\Delta ({u^2})u + V(x )u = h(u),{\text{ }}u > 0{\text{ in }}{\mathbb{R}^N}, \] 其中 $\varepsilon $ 是一个小的正参数，$N \ ge 3$ 和非线性$h$ 是临界增长。我们构建了上述问题的一系列正解 ${u_\varepsilon } \in {H^1}({\mathbb{R}^N})$，集中在 $V$ 的局部最小值附近作为 $\varepsilon \在 $h$ 的某些假设下为 0$。我们的结果在 $h(u) \sim \lambda {u^{q - 1}} + {u^{2 \cdot {2^ * } - 1}}{\text{ }}(2 0)$ 并完成了在一些最近的作品中所做的研究，从某种意义上说，在那些论文中，只有 $h(u) \sim \lambda {u^{q - 1}} + {u^{2 \cdot {2^ * } - 1}}{\text{ }}(4 0)$ 被考虑。此外，我们的主要结果还扩展了 Byeon 和 Jeanjean [14] 中使用的参数，这些参数处理具有亚临界非线性的薛定谔方程，到具有临界非线性的拟线性薛定谔方程。