Research article

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

  • 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.

    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

    Related Papers:

  • 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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