In this paper, we investigate the existence of $W_0^{1,1}(\Omega)$ solutions to the following elliptic equation with principal part having noncoercivity and singular quadratic term \begin{equation*} \left \{ \begin{array}{rl} -\text{div}\left(\frac{\nabla u}{(1+|u|)^{\gamma}}\right)+\frac{|\nabla u|^2}{u^{\theta}}=f,&x\in\Omega,\\ u=0,&x\in\partial\Omega, \end{array} \right. \end{equation*} where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N(N\geq3)$, $\gamma>0$, $\frac{N}{N-1}\leq\theta<2$, $f\in L^m(\Omega)(m\geq1)$ is a nonnegative function.

中文翻译：

---

具有奇异自然增长项的非线性椭圆方程的$ W_0 ^ {1,1}（\ Omega）$解的存在性

在本文中，我们研究以下椭圆方程的$ W_0 ^ {1,1}（\ Omega）$解的存在性，该椭圆方程的主要部分具有非矫顽力和奇异二次项\ begin {equation *} \ left \ {\ begin {数组} {rl}-\ text {div} \ left（\ frac {\ nabla u} {（1+ | u |）^ {\ gamma}} \ right）+ \ frac {| \ nabla u | ^ 2} {u ^ {\ theta}} = f，＆x \ in \ Omega，\\ u = 0，＆x \ in \ partial \ Omega，\ end {array} \ right。\ end {equation *}，其中$ \ Omega $是$ \ mathbb {R} ^ N（N \ geq3）$，$ \ gamma> 0 $，$ \ frac {N} {N-1}的有界光滑域\ leq \ theta <2 $，L ^ m（\ Omega）（m \ geq1）$中的$ f \是非负函数。