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

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

Keywords

• noncoercivity
• existence
• nonlinear elliptic equation