In this paper, we prove the existence of a solution of the strongly nonlinear degenerate p(x)-elliptic equation of type:where Ω is a bounded open subset of IR N , N ≥ 2, a is a Carathéodory function from Ω × IR × IR N into IR N , who satisfies assumptions of growth, ellipticity and strict monotonicity. The nonlinear term g: Ω × IR × IR N −→ IR checks assumptions of growth, sign condition and coercivity condition, while the right hand side f belongs to L 1 (Ω) .

In this article, we study the following degenerate unilateral problems:  $$ -\mbox{ div} (a(x,\nabla u))+H(x,u,\nabla u)=f,$$ which is subject to the Weighted Sobolev spaces with variable exponent $W^{1,p(x)}_{0}(\Omega,\omega)$, where $\omega$ is a weight function on $\Omega$, ($\omega$ is a measurable, a.e. strictly positive function on $\Omega$ and satisfying some integrability conditions). The function $H(x,s,\xi)$ is a nonlinear term satisfying some growth condition but no sign condition  and the right hand side $f\in L^1(\Omega)$.

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We consider, for a bounded open domain Ω in Rn; (n ≥ 1) and a function u : Ω → ℝm; (m ≥ 1) the quasilinear elliptic system: (QESw)(f,g) (0.1) Which is a Dirichlet problem. Here, v belongs to the dual space , f and g satisfy some stan- dard continuity and growth conditions. we will show the existence of a weak solution of this problem in the four following cases: σ is mono- tonic, σ is strictly monotonic, σ is quasi montone and σ derives from a convex potential.

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