Liouville theorems for quasi-harmonic functions

Let NN be a compact Riemannian manifold. A self-similar solution for the heat flow is a harmonic map from (Rn,e−|x|2/2(n−2)ds02) to NN (n≥3n≥3), which was also called a quasi-harmonic sphere (cf. Lin and Wang (1999) [1]). (Here ds02 is the Euclidean metric in Rn.) It arises from the blow-up analysis of the heat flow at a singular point. When N=R and without the energy constraint, we call this a quasi-harmonic function. In this paper, we prove that there is neither a nonconstant positive quasi-harmonic function nor a nonconstant Lp(Rn,e−|x|2/2(n−2)ds02)(p>nn−2) quasi-harmonic function. However, for all 1≤p≤n/(n−2)1≤p≤n/(n−2), there exists a nonconstant quasi-harmonic function in Lp(Rn,e−|x|2/2(n−2)ds02).