Regularity for mixed-order nonlinear fractional equations with degenerate coefficients

TL;DR

This work analyzes a nonlinear nonlocal equation with a leading operator formed by a weighted sum of fractional $p$‑Laplacians of orders $s$ and $t$ (with $0<s<t<1<p<\infty$) and possibly degenerate modulating coefficients $a(x,y),b(x,y)\ge0$, yielding a nonlocal double-phase structure. By developing Caccioppoli-type estimates, logarithmic lemmas, and an expansion of positivity framework, the authors establish local boundedness and Hölder regularity for weak solutions under natural kernel and coefficient assumptions; in the special case $a(\cdot,\cdot)\equiv1$, they prove a Harnack inequality that incorporates nonlocal tails. The analysis hinges on delicate tail estimates, the interplay between the two fractional phases, and the use of nonlocal Sobolev-type tools to control long-range interactions. The results extend regularity theory to a broad nonlocal double-phase setting and provide sharp, tail‑aware estimates that are relevant for nonautonomous fractional elliptic problems with mixed differentiability orders.

Abstract

We consider a class of nonlinear integro-differential equations whose leading operator is obtained as a superposition of $(-Δ_{p})^{s}$ and $(-Δ_{p})^{t}$, where $0<s<t<1<p<\infty$, weighted via two possibly degenerate coefficients $a(\cdot,\cdot),b(\cdot,\cdot) \ge 0$. 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 $s,t$, and $p$. Moreover, when $a(\cdot,\cdot) \equiv 1$, we also prove a Harnack inequality for weak solutions.

Theorems & Definitions (25)

• Remark 1.1
• Theorem 1.2: Local boundedness
• Theorem 1.3: Hölder regularity
• Theorem 1.4: Harnack inequality
• Remark 1.5
• Lemma 2.1: BOS
• Lemma 2.2: Ok23
• Lemma 2.3
• proof
• Lemma 2.4: MR1962933