Nonlinear degenerate elliptic equations in weighted Sobolev spaces

Abstract

We study the existence of solutions for the nonlinear degenerated elliptic problem $$\displaylines{ -\operatorname{div} a(x,u,\nabla u)=f \quad\text{in } \Omega,\cr u=0 \quad\text{on }\partial\Omega, }$$ where $\Omega$ is a bounded open set in $\mathbb{R}^N$, $N\geq2$, a is a Caratheodory function having degenerate coercivity $a(x,u,\nabla u)\nabla u\geq \nu(x)b(|u|)|\nabla u|^p$, 1<p<N, $\nu(\cdot)$ is the weight function, b is continuous and $f\in L^r(\Omega)$.

Keywords

• nonlinear degenerated elliptic operators
• weighted sobolev space
• monotony and rearrangement methods