Eigenfunctions for quasi-laplacian

Abstract

To study the regularity of heat flow, Lin-Wang[1] introduced the quasi-harmonic sphere, which is a harmonic map from $M=({\mathbb{R}}^m,e^{-\frac{|x{|}^2}{2(m-2)}}d{s}_0^2)$ to $N$ with finite energy. Here $d{s}_0^2$ is Euclidean metric in ${\mathbb{R}}^m$. Ding-Zhao [2] showed that if the target is a sphere, any equivariant quasi-harmonic spheres is discontinuous at infinity. The metric $g=e^{-\frac{|x{|}^2}{2(m-2)}}d{s}_0^2$ is quite singular at infinity and it is not complete. In this paper , we mainly study the eigenfunction of Quasi-Laplacian ${\Delta }_g=e^{\frac{|x{|}^2}{2(m-2)}}({\Delta }_{g_0}-{\nabla }_{g_0}h\cdot {\nabla }_{g_0})=e^{\frac{|x{|}^2}{2(m-2)}}{\Delta }_h$ for $h=\frac{|x{|}^2}{4}$. In particular, we show that non-constant eigenfunctions of ${\Delta }_g$ must be discontinuous at infinity and non-constant eigenfunctions of drifted Laplacian ${\Delta }_h={\Delta }_{g_0}-{\nabla }_{g_0}h\cdot {\nabla }_{g_0}$ is also discontinuous at infinity.