In this paper, we discuss the quasilinear Schrödinger equation with a Hardy potential $-\Delta u+V\left(x\right)u-\mu \frac{u}{{\left|x\right|}^2}-\Delta \left(u^2\right)u=g\left(u\right),x\in {\mathbb{R}}^N\setminus \left\{0\right\},$where $N\ge 3$, $0\le \mu <\bar{\mu }$=$\frac{{(N-2)}^2}{4}$, $\frac{1}{{\left|x\right|}^2}$ is called the Hardy potential. By using Jeanjean’s monotonicity trick, we admit a class of ground state solutions for the above problem, under the general Berestycki–Lions type assumptions on the nonlinearity $g$, as well as some weak assumptions on the potential $V$.

中文翻译：

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具有哈代势和 Berestycki-Lions 类型条件的拟线性薛定谔方程的存在基态解

在本文中，我们讨论了具有 Hardy 势的拟线性薛定谔方程 $-\Delta 你+伏\left(X\right)你-\mu \frac{你}{{\left|X\right|}^2}-\Delta \left({你}^2\right)你=G\left(你\right),X\in {\mathbb{电阻}}^N\setminus \left\{0\right\},$在哪里 $N\ge 3$, $0\le \mu <\bar{\mu }$=$\frac{{(N-2)}^2}{4}$, $\frac{1}{{\left|X\right|}^2}$称为哈代势。通过使用 Jeanjean 的单调性技巧，我们在非线性的一般 Berestycki-Lions 类型假设下承认上述问题的一类基态解$G$，以及一些关于潜力的弱假设 $伏$.