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

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.