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This article is devoted to study the existence of solutions for the strongly nonlinear \(p(x)\)-elliptic problem: \(- \Delta_{p(x)} (u) + \alpha_0 |u|^{p(x)-2}u = d(x)\frac{|\nabla u|^{p(x)}}{|u|^{p(x)}+1} + f- div g(x) \quad \text{in } \Omega, \) \(u \in W_0^{1,p(x)}(\Omega), \) Where \(\Omega\) is an open set of \(\mathbb{R}^N\), possibly of infinite measure, we will also give some…