We prove some existence results for nonlinear degenerated elliptic
problems of the form $$Au + g(x,u)= f-\mbox{div} F,$$ where $A(u) =
-\mbox{div}a(x,u,\nabla u)$ is a Leray-Lions, operator defined form
the weighted Sobolev space $W_0^{1,p}(\Omega,w)$ into its dual. The
right hand side, \ \ $f \in L^1(\Omega)$ and $ F \in {\displaystyle
\prod_{i=1}^N}L^{p'}(\Omega,w_i^*)$. Note that the Caracth\'eodory
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.