Quasilinear degenerated equations with $L^1$ datum and without coercivity in perturbation terms

Abstract

In this paper we study the existence of solutions for the generated boundary value problem, with initial datum being an element of $L^1(\Omega)+W^{-1, p'}(\Omega, w^{*})$ $$-{\rm div}a(x, u, \nabla u) + g(x, u, \nabla u) = f-{\rm div}F $$ where $a(.)$ is a Carathéodory function satisfying the classical condition of type Leray-Lions hypothesis, while $g(x, s, \xi)$ is a non-linear term which has a growth condition with respect to $\xi$ and no growth with respect to $s$, but it satisfies a sign condition on $s$.