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Existence of solutions for anisotropic quasilinear elliptic equations with variable exponent

Abstract We study the existence of solutions for a class of quasilinear elliptic equations involving the anisotropic -Laplace operator, on a bounded domain with smooth boundary. Since our differential operator involves partial derivatives with different variable exponents, we work on the anisotropic variable exponent Sobolev spaces. Using the Ekeland's variational principle and the mountain-pass theorem of Ambrosetti and Rabinowitz, we establish two existence results.