Entropy solutions to noncoercive nonlinear elliptic equations with measure data

Abstract

Let $\Omega\subseteq \mathbb{R}^N$ be a bounded domain. In this article, we investigate the existence of entropy solutions to the nonlinear elliptic problem $$\displaylines{ -\hbox{div}\Big(\frac{|\nabla u|^{(p-2)} \nabla u+c(x)u^\gamma}{(1+|u|)^{\theta(p-1)}}\big) +\frac{b(x)|\nabla u|^\lambda}{(1+|u|)^{\theta(p-1)}}=\mu,\quad x\in\Omega, \cr u(x)=0,\quad x\in \partial\Omega, }$$ where $\mu$ is a diffuse measure with bounded variation on $\Omega$, $0\leq\theta<1$ is a positive constants, 1<p<N, $0<\gamma\leq p-1$, $0<\lambda\leq p-1$, c(x) and b(x) belong to appropriate Lorentz spaces.

Keywords

• Entropy solution
• noncoercive
• nonlinear elliptic equations
• measure data