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In this paper, we prove the existence of a solution of the strongly nonlinear degenerate \(p(x)\)-elliptic equation of type: \(\mathcal{(P)}\left\{\begin{array}{rl} - div\; a(x,u,\nabla u) +g(x,u,\nabla u)& = f \quad in \;\Omega, \\ u = 0 \quad on \quad \partial\Omega, \end{array}\right.\) where \(\Omega\) is a bounded open subset of \( I\!\!R^{N}, N\geq 2, a\) is a…