This paper is concerned with the following quasilinear Schrödinger equation: $$-\Delta u+V(x)u-\frac{1}{2}\Delta (u^2)u= g(u), \quad x\in \mathbb{R}^N,$$ where $N\ge 3$, $V\in \mathcal{C}(\mathbb R^N,[0,\infty))$ and $g\in \mathcal{C}(\mathbb{R}, \mathbb{R})$ is superlinear at infinity. By using variational and some new analytic techniques, we prove the above problem admits ground state solutions under mild assumptions on $V$ and $g$. Moreover, we establish a minimax characterization of the ground state energy. Especially, we impose some new conditions on $V$ and more general assumptions on $g$. For this, some new tricks are introduced to overcome the competing effect between the quasilinear term and the superlinear reaction. Hence our results improve and extend recent theorems in several directions.

中文翻译：

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具有可变势和超线性反应的拟线性Schrödinger方程的基态解

本文关注以下准线性Schrödinger方程：$$-\ Delta u + V（x）u- \ frac {1} {2} \ Delta（u ^ 2）u = g（u），\ quad x \ \ mathbb {R} ^ N，$$，其中$ N \ ge 3 $，$ V \ in \ mathcal {C}（\ mathbb R ^ N，[0，\ infty））$和$ g \ in \ mathcal {C}（\ mathbb {R}，\ mathbb {R}）$在无穷大处是超线性的。通过使用变分和一些新的分析技术，我们证明了上述问题在对$ V $和$ g $的温和假设下可以接受基态解。此外，我们建立了基态能量的极小极大特征。特别是，我们对$ V $施加了一些新条件，并对$ g $施加了更一般的假设。为此，引入了一些新的技巧来克服准线性项和超线性反应之间的竞争效应。因此，我们的结果在几个方向上改进和扩展了最新定理。