Higher Hölder regularity for fractional $(p,q)$-Laplace equations

TL;DR

The paper analyzes local Hölder regularity for the nonlocal mixed-order equation $(-\Delta_p)^s u + (-\Delta_q)^t u = 0$ with $s,t\in(0,1)$ and $p,q>1$, establishing Hölder continuity with an explicit exponent below the threshold $\Gamma = \min\big(1, \frac{sp}{p-1}\big)$ (under the regime $\frac{sp}{p-1} \ge \frac{tq}{q-1}$). The authors develop a three-case framework (Case I: $1<q\leq 2 \leq p$, Case II: $1<p\leq 2 \leq q$, Case III: $p,q\in(1,2]$) and combine Besov-type regularity improvements with a Moser-type iteration on difference quotients to derive higher Hölder regularity and a Liouville-type theorem for globally bounded solutions. A key technical contribution is the refined control of second-order difference quotients in Besov spaces, propagated through iterative schemes to achieve explicit Hölder exponents that depend on $s,p,t,q$ and tail data. The results extend the regularity theory for the fractional $p$-Laplacian to mixed-order nonlocal operators and provide quantitative a priori bounds that depend on tails and the parameters, with potential implications for nonlinear nonlocal PDEs and Liouville-type analyses.

Abstract

We study the fractional $(p,q)$-Laplace equation $$ (-Δ_p)^s u +(-Δ_q)^t u= 0 $$ for $s,t\in(0,1)$ and $p,q\in(1,\infty)$. We establish Hölder estimates with an explicit exponent. As a consequence, we derive a Liouville-type theorem. Our approach builds on techniques previously developed for the fractional $p$-Laplace equation, relying on a Moser-type iteration for difference quotients.

Theorems & Definitions (36)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Corollary 1.4
• proof
• Lemma 2.1
• Theorem 2.2
• Definition 2.3
• Proposition 3.1
• proof