Entropy solutions of nonlinear elliptic equations with measurable boundary conditions and without strict monotonocity conditions

We prove some existence results for nonlinear degenerate elliptic problems of the form Au+ g(x, u) = f - divF, where $A(u) = -diva(x, u, \nabla u)$ is a Leray-Lions, operator defined form the weighted Sobolev space $W^{1,p}_0 (\Omega,w)$ into its dual. The right hand side, $f \in L^1(\Omega)$ and $F \in \prod\limits_{i=1}^{N} L^{p'}(\Omega,\omega^*_i)$. Note that the Carathéodory function $a(x, s,\xi)$ satisfies only the large monotonicity instead of the monotonicity strict condition. We overcome this difficulty by using the $L^1$-version of Minty's lemma.