Eigenfunctions for quasi-Laplacian

To study the regularity of heat flow, Lin and Wang (1999) introduced the quasi-harmonic sphere, which is a harmonic map from M = (R-m, e(-vertical bar x vertical bar 2/2(m - 2)) ds(0)(2)) to N with finite energy. Here ds(0)(2) is Euclidean metric in R-m. Ding and Yongqiang (2006) showed that if the target is a sphere, any equivariant quasi-harmonic spheres is discontinuous at infinity. The metric g = e(-vertical bar x vertical bar 2/2(m) (- 2)) ds(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(vertical bar x vertical bar 2/2(m - 2)) (Delta(g0) - del(g0)h . del(g0)) = e(vertical bar x vertical bar 2/2(m - 2)) Delta(h) for h = vertical bar x vertical bar(2)/4. In particular, we show that there are infinite number of eigenvalues (of the quasiLaplacian Delta(g)) of which the corresponding eigenfunctions are discontinuous at infinity and any non-constant eigenfunction of drifted Laplacian Delta(h) = Delta g(0) - del(g0)h . del(g0) is discontinuous at infinity. (C) 2019 Elsevier Ltd. All rights reserved.