Abstract. In this work, we prove the existence of a weak solution of an anisotropic parabolic problem with measure data u t + Au + F(u,Du) = μ and u(0) = μ 0 with μ and μ 0 two Radon bounded measures. The operator A is a Leray-Lions operator with anisotropic growth conditions. Our approach is based on the anisotropic Sobolev inequality, a regularity result, a compactness result, and an integration by parts formula.Mathematics subject classification (2010): 28C05, 35K10, 35K20, 35K55.

In this work, we prove the existence of weak solution for a class of anisotropic parabolic problems with Orlicz data. Our approach is based on the anisotropic Sobolev inequality, a smoothness, and compactness results.

In this paper we prove existence results for distributional solutions of nonlinear elliptic systems with a measure data. The functional setting involves Lebesgue-Sobolev spaces as well as weak Lebesgue (Marcinkiewicz) spaces with variable exponents W01,p(·)(Ω) and Mp(·)(Ω) respectively.

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In this paper, we prove existence and regularity results for weak solutions in the framework of anisotropic Sobolev spaces for a class of nonlinear anisotropic elliptic equations in the whole double-struckRN with variable exponents and locally integrable data. Our approach is based on the anisotropic Sobolev inequality, a smoothness, and compactness results. The functional setting involves Lebesgue–Sobolev spaces with variable exponents. Copyright © 2016 John Wiley & Sons, Ltd.