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

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 $n\ge 2$, $\sigma \in (0,1)$, 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 $u(x)\le C|x{|}^{-(n-2\sigma )/2}$ 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 ${\mathbb{B}}^{n+1}$.

34 pages. Fixed some typos. arXiv admin note: text overlap with arXiv:2009.02069

References in corpus (3)

• On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions
• On a fractional Nirenberg problem, part II: existence of solutions
• Local estimates for conformal $Q$-curvature equations