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Regularity for quasi-linear degenerate elliptic equations with VMO coefficients

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Abstract

In this paper we establish an interior regularity of weak solution for quasi-linear degenerate elliptic equations under the subcritical growth if its coefficient matrix A(x, u) satisfies a VMO condition in the variable x uniformly with respect to all u, and the lower order item B(x, u, ▽u) satisfies the subcritical growth (1.2). In particular, when F(x) ∈ L q(Ω) and f(x) ∈ L γ(Ω) with q, γ > max {p, n} for any 1 < p < + ∞, we obtain interior Hölder continuity of any weak solution of (1.1) u ∈ C 0,κloc (Ω) with an index κ = min {1 − n/q, 1 − n/γ}.

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  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, P. R. China

    Shen Zhou Zheng

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  1. Shen Zhou Zheng
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Correspondence to Shen Zhou Zheng.

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Supported by National Natural Science Foundation of China (No. 10671022)

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Zheng, S.Z. Regularity for quasi-linear degenerate elliptic equations with VMO coefficients. Acta. Math. Sin.-English Ser. 24, 1909–1924 (2008). https://doi.org/10.1007/s10114-008-5644-3

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