We establish the existence of a non-trivial weak solution to the following singular quasilinear equation with Hardy potential and singular quadratic gradient term:$\{\begin{array}{cc}\hfill & -\mathrm{\Delta }u-\mu \frac{u}{|x{|}^2}=\frac{|\mathrm{\nabla }u{|}^2}{u}+f(x,u)\text{in}\mathrm{\Omega },\hfill \\ \hfill & u>0\text{in}\mathrm{\Omega },\hfill \\ \hfill & u=0\text{on}\partial \mathrm{\Omega },\hfill \end{array}$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain, $\mu >0,0\in \mathrm{\Omega },N\ge 3$. We show that there exists a solution $u\in H_0^1(\mathrm{\Omega })$ to the above problem. The notable characteristic of this problem is that it includes quadratic gradient nonlinearity and strong singularity.

中文翻译：

---

具有二次梯度项的奇异椭圆方程

我们建立了以下具有Hardy势和奇异二次梯度项的奇异拟线性方程的非平凡弱解的存在：$\{\begin{array}{cc}\hfill & -\mathrm{\Delta }ü-\mu \frac{ü}{|X{|}^{\mathrm{2个}}}=\frac{|\mathrm{\nabla }ü{|}^{\mathrm{2个}}}{ü}+F（X，ü）\text{在}\mathrm{\Omega }，\hfill \\ \hfill & ü>0\text{在}\mathrm{\Omega }，\hfill \\ \hfill & ü=0\text{上}\partial \mathrm{\Omega }，\hfill \end{array}$ 在哪里 $\mathrm{\Omega }\subset {\mathbb{[R}}^{ñ}$ 是一个光滑的有界域， $\mu >0，0\in \mathrm{\Omega }，ñ\ge 3$。我们表明存在解决方案$ü\in H_0^{\mathrm{1个}}（\mathrm{\Omega }）$以上问题。这个问题的显着特征是它包括二次梯度非线性和强奇异性。