Existence of $W_0^{1,1}(\Omega)$ solutions to nonlinear elliptic equation with singular natural growth term

Maoji Ri; Shuibo Huang; Qiaoyu Tian; Zhan-Ping Ma; Maoji Ri; Shuibo Huang; Qiaoyu Tian; Zhan-Ping Ma

doi:10.3934/math.2020371

AIMS Mathematics

2020, Volume 5, Issue 6: 5791-5800. doi: 10.3934/math.2020371

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Research article

Existence of $W_0^{1,1}(\Omega)$ solutions to nonlinear elliptic equation with singular natural growth term

1.
School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou, Gansu 730030, P. R. China
2.
Key Laboratory of Streaming Data Computing Technologies and Application, Northwest Minzu University, Lanzhou, Gansu 730030, P. R. China
3.
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454003 Henan, P. R. China

Received: 06 April 2020 Accepted: 05 July 2020 Published: 13 July 2020
MSC : 35J67, 35R10

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 $\mathbb{R}^N(N\geq3)$, $\gamma \gt 0$, $\frac{N}{N-1}\leq\theta \lt 2$, $f\in L^m(\Omega)(m\geq1)$ is a nonnegative function.
- noncoercivity,
- existence,
- nonlinear elliptic equation
Citation: Maoji Ri, Shuibo Huang, Qiaoyu Tian, Zhan-Ping Ma. Existence of $W_0^{1,1}(\Omega)$ solutions to nonlinear elliptic equation with singular natural growth term[J]. AIMS Mathematics, 2020, 5(6): 5791-5800. doi: 10.3934/math.2020371

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Abstract

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 $\mathbb{R}^N(N\geq3)$, $\gamma \gt 0$, $\frac{N}{N-1}\leq\theta \lt 2$, $f\in L^m(\Omega)(m\geq1)$ is a nonnegative function.

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