A non-compactness result on the fractional Yamabe problem in large dimensions

Abstract

Let $(X^{n+1},g^{+})$ be an $(n+1)$-dimensional asymptotically hyperbolic manifold with a conformal infinity $(M^n,[\hat{h}])$. The fractional Yamabe problem addresses to solve \[P^γ[g^+,\hat{h}] (u) = cu^{n+2γ\over n-2γ}, \quad u > 0 \quad \text{on } M\] where $c\in \mathbb{R}$ and $P^{\gamma }[g^{+},\hat{h}]$ is the fractional conformal Laplacian whose principal symbol is $(-\Delta {)}^{\gamma }$. In this paper, we construct a metric on the half space $X={\mathbb{R}}_{+}^{n+1}$, which is conformally equivalent to the unit ball, for which the solution set of the fractional Yamabe equation is non-compact provided that $n\ge 24$ for $\gamma \in (0,{\gamma }^{\ast })$ and $n\ge 25$ for $\gamma \in [{\gamma }^{\ast },1)$ where ${\gamma }^{\ast }\in (0,1)$ is a certain transition exponent. The value of ${\gamma }^{\ast }$ turns out to be approximately 0.940197.

48 pages. Introduction and some part of the proof are updated

References in corpus (6)

• On higher order extensions for the fractional Laplacian
• Compactness of conformal metrics with constant $Q$-curvature. I
• Further results on the fractional Yamabe problem: the umbilic case
• On perturbations of the fractional Yamabe problem
• Bubbling solutions for nonlocal elliptic problems
• Existence results for the fractional Nirenberg problem

Cited by in corpus (5)

• On perturbations of the fractional Yamabe problem
• On local behavior of singular positive solutions to nonlocal elliptic equations
• Fractional nonlinear Schrödinger equations with singular potential in ${\mathbf{R}}^n$
• Compactness of solutions to nonlocal elliptic equations
• Classification of finite energy solutions to the fractional Lane-Emden-Fowler equations with slightly subcritical exponents