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In this article, we deal with the existence of solutions for a class of nonlinear elliptic system with-Laplacian operator. The work is based on the Leray–Schauder principle with the use of the topological degree of Berkovits applied to an abstract Hammerstein equation associated to our system and presenting a certain lack of compactness. In this study, we establish existence results when the source terms depend on the gradient which are convection terms.

On a nonlinear elliptic system involving the ( p ( x ), q ( x ))-Laplacian operator with gradient dependence