Existence and Boundedness of Solutions for Elliptic Equations in General Domains

This article is devoted to study the existence of solutions for the strongly nonlinear p(x)p(x)-elliptic problem: - \Delta_{p(x)} (u) + \alpha_0 |u|^{p(x)-2}u = d(x)\frac{|\nabla u|^{p(x)}}{|u|^{p(x)}+1} + f- div g(x) \quad \text{in } \Omega, −Δ p(x) ​ (u)+α 0 ​ ∣u∣ p(x)−2 u=d(x) ∣u∣ p(x) +1 ∣∇u∣ p(x) ​ +f−divg(x)in Ω, u \in W_0^{1,p(x)}(\Omega), u∈W 0 1,p(x) ​ (Ω), Where \OmegaΩ is an open set of \mathbb{R}^NR N , possibly of infinite measure, also we will give some regularity results for these solutions