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Harnack’s Inequality and Applications of Quasilinear Degenerate Elliptic Equations with Rough and Singular Coefficients

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Abstract

We consider the quasilinear degenerate elliptic equation with rough and singular coefficients of the form

$$\begin{aligned} div\left( {A(x,u,\nabla u)} \right) = B(x,u,\nabla u) \end{aligned}$$

in an open set \(\Omega \subset {\mathbb {R}^n},\) where the quadratic form associated with the principle part of this equation allows vanishing and the coefficients in structural conditions on \(A(x,u,\nabla u)\) and \(B(x,u,\nabla u)\) require no smoothness but belong to some Stummel–Kato class. We prove a Fefferman–Phong type inequality related to the Stummel–Kato class and then an embedding inequality. Based on these inequalities, the local boundedness and Harnack’s inequality of the weak solutions are derived. As applications, the continuity and Hölder continuity for the nonnegative weak solutions are given.

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Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710129, People’s Republic of China

    Pengcheng Niu & Leyun Wu

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  1. Pengcheng Niu
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  2. Leyun Wu
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Correspondence to Leyun Wu.

Additional information

This work is supported by the National Natural Science Foundation of China (No. 11771354) and the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2016JM2013).

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Niu, P., Wu, L. Harnack’s Inequality and Applications of Quasilinear Degenerate Elliptic Equations with Rough and Singular Coefficients. Bull Braz Math Soc, New Series 49, 279–312 (2018). https://doi.org/10.1007/s00574-017-0054-8

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  • DOI: https://doi.org/10.1007/s00574-017-0054-8

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