A priori estimates versus arbitrarily large solutions for fractional semi-linear elliptic equations with critical Sobolev exponent

TL;DR

This paper investigates the behavior of positive solutions to fractional semi-linear elliptic equations with critical Sobolev exponent near isolated singularities, revealing dimension-dependent estimates and constructions of arbitrarily large solutions.

Contribution

It establishes a priori bounds in lower dimensions and constructs examples of unbounded solutions in higher dimensions for fractional elliptic equations with critical exponent.

Findings

Solutions are bounded near the singularity in lower dimensions.

In higher dimensions, solutions can be arbitrarily large near the singularity.

Results extend to prescribed boundary mean curvature problems.

Abstract

We study positive solutions to the fractional semi-linear elliptic equation $$ (- \Delta)^\sigma u = K(x) u^\frac{n + 2 \sigma}{n - 2 \sigma} ~~~~~~ in ~ B_2 \setminus \{ 0 \} $$ with an isolated singularity at the origin, where $K$ is a positive function on $B_2$, the punctured ball $B_2 \setminus \{ 0 \} \subset \mathbb{R}^n$ with $n \geq 2$, $\sigma \in (0, 1)$, and $(- \Delta)^\sigma$ is the fractional Laplacian. In lower dimensions, we show that, for any $K \in C^1 (B_2)$, a positive solution $u$ always satisfies that $u(x) \leq C |x|^{ - (n - 2 \sigma)/2 }$ near the origin. In contrast, we construct positive functions $K \in C^1 (B_2)$ in higher dimensions such that a positive solution $u$ could be arbitrarily large near the origin. In particular, these results also apply to the prescribed boundary mean curvature equations on $\mathbb{B}^{n+1}$.