Stable solutions to p-Laplacian type equations with variable coefficients

Abstract

We study stable regular solutions of the variable-coefficient equation −div((A(x)∇ u·∇ u)^p/2−1A(x)∇ u)=f(u) in B₁⊂ℝⁿ, where p>1, f∈ C¹(ℝ) is nonnegative, and A is a locally Lipschitz, symmetric, uniformly elliptic matrix field. We prove quantitative interior Hölder estimates for n<p+4p/(p−1) when p≥2, and for n<5p when 1<p<2. We also obtain interior energy estimates and higher integrability of the gradient in every dimension. These estimates are given in terms of |u|_L¹(B₁), with constants depending only on n,p, the ellipticity bounds, and a local L^∞ bound for DA, but not on f or higher derivatives of A. These results extend the corresponding interior estimates of Cabré, Miraglio, and Sanchón for the p-Laplace equation to this variable-coefficient setting. The proof uses weighted Hessian estimates to control coefficient errors and a comparison between radial and full gradient energies.

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BibTeX RIS

BibTeX

@misc{luo2026stable,
  title = {{Stable solutions to $p$-Laplacian type equations with variable coefficients}},
  author = {Weican Luo},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00730},
  eprint = {2611.00730},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00730},
}

RIS

TY  - UNPB
TI  - Stable solutions to p-Laplacian type equations with variable coefficients
AU  - Weican Luo
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00730
UR  - https://arxiv.org/abs/2611.00730
AB  - We study stable regular solutions of the variable-coefficient equation −div((A(x)∇ u·∇ u)^p/2−1A(x)∇ u)=f(u) in B_1⊂ℝ^n, where p>1, f∈ C^1(ℝ) is nonnegative, and A is a locally Lipschitz, symmetric, uniformly elliptic matrix field. We prove quantitative interior Hölder estimates for n<p+4p/(p−1) when p≥2, and for n<5p when 1<p<2. We also obtain interior energy estimates and higher integrability of the gradient in every dimension. These estimates are given in terms of |u|_L^1(B_1), with constants depending only on n,p, the ellipticity bounds, and a local L^∞ bound for DA, but not on f or higher derivatives of A. These results extend the corresponding interior estimates of Cabré, Miraglio, and Sanchón for the p-Laplace equation to this variable-coefficient setting. The proof uses weighted Hessian estimates to control coefficient errors and a comparison between radial and full gradient energies.
ER  -

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