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In this work, we shall be concerned with the existence of weak solutions of anisotropic elliptic operators \(Au +\sum_{i=1}^{N}g_{i}(x, u, \nabla u)+\sum_{i=1}^{N}H_{i}(x, \nabla u)=f-\sum_{i=1}^{N} \frac{\partial }{\partial x_{i}}k_{i}\) where the right hand side \(f\) belongs to \(L^{p^{'}_{\infty}}(\Omega)\) and \(k_{i}\) belongs to \(L^{p_{i}^{'}}(\Omega)\) for \(i=1,...,N\) and \(A\) is a Leray-Lions operator. The critical growth condition on \(g_{i}\) is…

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…