Existence of renormalized solutions for some degenerate and non-coercive elliptic equations

Abstract

This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by \alignat2&-{\rm div}( b(|u|)|\nabla u|^{p-2}\nabla u) + d(|u|)|\nabla u|^p = f - {\rm div}(c(x)|u|^{\alpha}) &\quad&in \Omega, & u = 0 &\quad&on \partial\Omega, where $\Omega$ is a bounded open set of $\mathbb{R}^N$ ($N\geq2$) with $1<p<N$ and $f \in L^1(\Omega),$ under some growth conditions on the function $b(\cdot)$ and $d(\cdot),$ where $c(\cdot)$ is assumed to be in $L^{\frc{N}{(p-1)}}(\Omega).$ We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.

Keywords

• renormalized solution
• nonlinear elliptic equation
• non-coercive problem