Jing Yu, Jun Zheng, Regularity of solutions for double-phase elliptic equations involving measures

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DOI: 10.23952/jnva.10.2026.4.04

Volume 10, Issue 4, 1 August 2026, Pages 779-798

Abstract. In this paper, we study regularity for the double-phase problem

$-\mathrm{div}\left( a(x)|\nabla u|^{p-2}\nabla u + |\nabla u|^{q-2}\nabla u \right) = \mu$ in $\mathcal{D}'(\Omega),$

where $p$ and $q$ are positive constants satisfying $p > q \geq 2$ , $a(x)$ is locally Hölder continuous and has positive lower and upper bounds, $\mu$ is a nonnegative Radon measure satisfying $\mu(B_s) \leq Cs^m$ with some constant $C > 0$ for all balls $\overline{B}_s \subset \Omega$ , $\Omega$ is a bounded domain in $\mathbb{R}^n (n \geq 2)$ , and $m\geq n-1$ is a constant. For different $m$ , we prove Hölder continuity with different exponents for the locally bounded weak solutions, as well as their gradients by using the De Giorgi-Nash-Moser iteration and a freezing coefficient argument.

How to Cite this Article:
J. Yu, J. Zheng, Regularity of solutions for double-phase elliptic equations involving measures, J. Nonlinear Var. Anal. 10 (2026), 779-798.