Regularity for mixed-order nonlinear fractional equations with degenerate coefficients

Ho-Sik Lee; Jihoon Ok; Kyeong Song; Ho-Sik Lee; Jihoon Ok; Kyeong Song

doi:10.3934/mine.2026007

Mathematics in Engineering

2026, Volume 8, Issue 2: 181-208. doi: 10.3934/mine.2026007

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Regularity for mixed-order nonlinear fractional equations with degenerate coefficients

1.
Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany
2.
Department of Mathematics, Institute for Mathematical and Data Science, Sogang University, Seoul 04107, Republic of Korea
3.
School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea

Received: 08 December 2025 Revised: 02 March 2026 Accepted: 14 March 2026 Published: 19 March 2026

We consider a class of nonlinear integro-differential equations whose leading operator is modeled on a superposition of $ (-\Delta_{p})^{s} $ and $ (-\Delta_{p})^{t} $, where $ 0 < s < t < 1 < p < \infty $, weighted via two possibly degenerate coefficients $ a(\cdot, \cdot) \ge 0 $ and $ b(\cdot, \cdot) \ge 0 $, respectively. We prove local boundedness and Hölder regularity of its weak solutions under natural assumptions on the coefficients $ a(\cdot, \cdot) $, $ b(\cdot, \cdot) $ and the powers $ s $, $ t $, $ p $. Moreover, when $ a(\cdot, \cdot) \equiv 1 $, we also prove a Harnack inequality for weak solutions.
- nonlinear nonlocal operator,
- mixed order,
- degenerate coefficient,
- regularity,
- Harnack inequality
Citation: Ho-Sik Lee, Jihoon Ok, Kyeong Song. Regularity for mixed-order nonlinear fractional equations with degenerate coefficients[J]. Mathematics in Engineering, 2026, 8(2): 181-208. doi: 10.3934/mine.2026007

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Abstract

We consider a class of nonlinear integro-differential equations whose leading operator is modeled on a superposition of $ (-\Delta_{p})^{s} $ and $ (-\Delta_{p})^{t} $, where $ 0 < s < t < 1 < p < \infty $, weighted via two possibly degenerate coefficients $ a(\cdot, \cdot) \ge 0 $ and $ b(\cdot, \cdot) \ge 0 $, respectively. We prove local boundedness and Hölder regularity of its weak solutions under natural assumptions on the coefficients $ a(\cdot, \cdot) $, $ b(\cdot, \cdot) $ and the powers $ s $, $ t $, $ p $. Moreover, when $ a(\cdot, \cdot) \equiv 1 $, we also prove a Harnack inequality for weak solutions.

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