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Existence of $W_0^{1,1}(\Omega)$ solutions to nonlinear elliptic equation with singular natural growth term
AIMS Mathematics ( IF 1.8 ) Pub Date : 2020-07-13 , DOI: 10.3934/math.2020371
Maoji Ri , , Shuibo Huang , Qiaoyu Tian , Zhan-Ping Ma , ,

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。
更新日期:2020-07-20
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