Eigenfunctions for quasi-laplacian

To study the regularity of heat flow, Lin-Wang[1] introduced the quasi-harmonic sphere, which is a harmonic map from $M=({\mathbb{R}}^m,e^{-\frac{\mathrm{\mid }x{\mathrm{\mid }}^2}{2(m-2)}}d{s}_0^2)$ 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 $g=e^{-\frac{\mathrm{\mid }x{\mathrm{\mid }}^2}{2(m-2)}}d{s}_0^2$ 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 ${\mathrm{\Delta }}_g$ 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.