Refined regularity for nonlocal elliptic equations and applications

In this paper, we establish refined regularity estimates for nonnegative solutions to the fractional Poisson equation $$(-\Delta)^s u(x) =f(x),\,\, x\in B_1(0).$$ Specifically, we have derived H\"{o}lder, 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.