Existence of solutions for degenerated problems in $L^1$ having lower order terms with natural growth

Abstract

We prove the existence of a solution for a strongly nonlinear degenerated problem associated to the equation

where $A$ is a Leray–Lions operator from the weighted Sobolev space $W_0^{1,p}(\Omega ,w)$ into its dual $W^{-1,p^{\prime }}(\Omega ,w^{\ast })$. While $g(x,s,\xi )$ is a nonlinear term having natural growth with respect to $\xi$ and no growth with respect to $s$, it satisfies a sign condition on $s$, i.e., $g(x,s,\xi )\cdot s\ge 0$ for every $s\in R$. The right-hand side $f$ belongs to $L^1(\Omega )$.