A priori estimates versus arbitrarily large solutions for fractional semilinear elliptic equations with critical Sobolev exponent
Abstract
We study positive solutions to the fractional semilinear 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 $\mathbb{B}^{n+1}$.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.09048
 Bibcode:
 2021arXiv211009048D
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 34 pages