On the density and multiplicity of solutions to the fractional Nirenberg problem

TL;DR

This paper proves the existence of infinitely many solutions to the fractional Nirenberg problem on spheres and related equations in Euclidean space, showing the density of smooth conformal metrics and analyzing bubbling phenomena.

Contribution

It introduces a variational gluing method and detailed blow-up analysis to establish multiple solutions and density results for fractional curvature problems.

Findings

Existence of infinitely many multi-bump solutions.

Density of smooth conformal metrics in the $C^0$ topology.

Infinitely many solutions for fractional Laplacian equations with asymptotically periodic $K(x)$.

Abstract

This paper is devoted to establishing some results on the density and multiplicity of solutions to the fractional Nirenberg problem which is equivalent to studying the conformally invariant equation $P_\sigma(v)=K v^{\frac{n+2\sigma}{n-2\sigma}}$ on the standard unit sphere $(\mathbb{S}^n,g_0)$ with $\sigma\in (0,1)$ and $n\geq 2$, where $P_\sigma$ is the intertwining operator of order $2\sigma$ and $K$ is the prescribed curvature function. More specifically, by using the variational gluing method, refined analysis of bubbling behavior, extension formula, as well as the blow up analysis arguments, we obtain the existence of infinitely many multi-bump solutions. In particular, we show the smooth curvature functions of metrics conformal to $g_0$ are dense in the $C^{0}$ topology. Moreover, the related fractional Laplacian equations $(-\Delta)^{\sigma} u=K(x) u^{\frac{n+2\sigma}{n-2\sigma}}$ in $\mathbb{R}^n$, with $K(x)$ being asymptotically periodic in one of the variables, are also studied and infinitely many solutions are obtained under natural flatness assumptions.