Uniqueness and Symmetry Breaking for Singular $Q$-curvature Equations in $\R^n$

Abstract

For $n\ge3$ and $\gamma>-1$, we study normal solutions of \begin{equation*} (-\Delta)^{n/2}u=(n-1)!\,|x|^{n\gamma}e^{nu} \quad\hbox{in }\R^n. \end{equation*} We reformulate this problem on the cylinder via the Emden--Fowler transform. We first establish Kelvin symmetry for normal solutions. We then prove existence and uniqueness, up to dilation, of the radial normal solution and establish its radial nondegeneracy. Next, we carry out a detailed spectral analysis of the linearized operator and construct branches of nonradial normal solutions by the Crandall--Rabinowitz theorem. Finally, we prove that every normal solution is radial when $\gamma>0$ is sufficiently small.