Refined regularity for nonlocal elliptic equations and applications

TL;DR

This paper develops refined local regularity theory for nonnegative solutions of the fractional Poisson equation $(-\Delta)^s u=f$ in $B_1$, proving Hölder, Schauder, and Ln-Lipschitz estimates under only local $L^\infty$ bounds on $u$ and $f$. The key technique is a direct, non-extension analysis that splits $u$ into an $s$-harmonic part and a Newtonian-potential part, enabling precise control of higher derivatives via the Poisson kernel and its derivatives. The results provide sharp regularity across regimes of $2s+\alpha$, including Schauder when $2s+\alpha\notin\mathbb{N}$ and Ln-Lipschitz or $C^{2s+\alpha}$ with Dini data when $2s+\alpha\in\mathbb{N}$, all with estimates depending only on local norms. As applications, the authors derive singularity and decay estimates for fractional Lane-Emden type equations in unbounded domains and obtain a priori bounds via an auxiliary-function approach that avoids the doubling lemma, thereby enhancing tools for qualitative analysis of nonlinear fractional elliptic problems in unbounded settings.

Abstract

In this paper, we establish refined regularity estimates for nonnegative solutions to the fractional Poisson equation $$ (-Δ)^s u(x) =f(x),\,\, x\in B_1(0). $$ Specifically, we have derived Hölder, Schauder, and Ln-Lipschitz regularity estimates for any nonnegative solution $u,$ provided that only the local $L^\infty$ norm of $u$ is bounded. These estimates stand in sharp contrast to the existing results where the global $L^\infty$ norm of $u$ is required. Our findings indicate that the local values of the solution $u$ and $f$ are sufficient to control the local values of higher order derivatives of $u$. Notably, this makes it possible to establish a priori estimates in unbounded domains by using blowing up and re-scaling argument. As applications, we derive singularity and decay estimates for solutions to some super-linear nonlocal problems in unbounded domains, and in particular, we obtain a priori estimates for a family of fractional Lane-Emden type equations in $\mathbb{R}^n.$ This is achieved by adopting a different method using auxiliary functions, which is applicable to both local and nonlocal problems.

Theorems & Definitions (22)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 1.10