LAUSR

We consider the degenerate nonlinear elliptic equation (E) : $${\mathcal {A}}(u)= g-{\text {div}}(f)$$A(u)=g-div(f), where $${\mathcal {A}}(u)=-{\text {div}}(a(x,u,\nabla u))$$A(u)=-div(a(x,u,∇u)) is a Leray-Lions operator defined on $$W_0^{1,p(\cdot )}(\Omega )$$W01,p(·)(Ω) allowed to be non linear degenerated. The main gaol of this paper is to prove in first, an $$L^{\infty }(\Omega )$$L∞(Ω) estimate for the bounded solution of (E), and then the existence of a weak and a renormalized solution of (E), with $$f\in (L^{r(\cdot )}(\Omega ))^N, g\in L^{q(\cdot )}(\Omega )$$f∈(Lr(·)(Ω))N,g∈Lq(·)(Ω), where $$r(\cdot )$$r(·) and $$q(\cdot )$$q(·) satisfies the following conditions : $$\begin{aligned} {\left\{ \begin{array}{ll} r(x)>\frac{N}{p(x)-1}, r(x)\ge p'(x)&{}\quad \forall x \in \Omega ,\\ q(x)>\max \left( \frac{N}{p(x)},1\right) , q(x)\ge p'(x)&{}\quad \forall x \in \Omega . \end{array}\right. } \end{aligned}$$r(x)>Np(x)-1,r(x)≥p′(x)∀x∈Ω,q(x)>maxNp(x),1,q(x)≥p′(x)∀x∈Ω.