Existence of a weak bounded solution for nonlinear degenerate elliptic equations in Musielak-Orlicz spaces

In this paper, we show the existence of solutions for the nonlinear elliptic equations of the form

{-div a(x,u,∇u)=f,u∈W01Lϕ(Ω)∩L∞(Ω),\left\{ {\matrix{ { - {\rm{div}}\,a\left( {x,u,\nabla u} \right) = f,} \hfill \cr {u \in W_0^1L\varphi \left( \Omega \right) \cap {L^\infty }\left( \Omega \right),} \hfill \cr } } \right.

where a(x,s,ξ)⋅ξ≥ϕ¯x-1(ϕ(x,h(|s|)))ϕ(x,|ξ|)a\left( {x,s,\xi } \right) \cdot \xi \ge \bar \varphi _x^{ - 1}\left( {\varphi \left( {x,h\left( {\left| s \right|} \right)} \right)} \right)\varphi \left( {x,\left| \xi \right|} \right) and h : ℝ+→]0, 1] is a continuous decreasing function with unbounded primitive. The second term f belongs to L^N(Ω) or to L^m(Ω), with m=rNr+1m = {{rN} \over {r + 1}} for some r > 0 and φ is a Musielak function satisfying the Δ₂-condition.