A remark on a result of Ding-Jost-Li-Wang

Abstract

Let ( M , g ) (M,g) be a compact Riemannian surface without boundary, W 1 , 2 ( M ) W^{1,2}(M) be the usual Sobolev space, and J : W 1 , 2 ( M ) → R J: W^{1,2}(M)\rightarrow \mathbb {R} be the functional defined by \[ J ( u ) = 1 2 ∫ M | ∇ u | 2 d v g + 8 π ∫ M u d v g − 8 π log ⁡ ∫ M h e u d v g , J(u)=\frac {1}{2}\int _M|\nabla u|^2dv_g+8\pi \int _M udv_g-8\pi \log \int _Mhe^udv_g, \] where h h is a positive smooth function on M M . In an inspiring work (Asian J. Math., vol. 1, pp. 230-248, 1997), Ding, Jost, Li and Wang obtained a sufficient condition under which J J achieves its minimum. In this note, we prove that if the smooth function h h satisfies h ≥ 0 h\geq 0 and h ≢ 0 h\not \equiv 0 , then the above result still holds. Our method is to exclude blow-up points on the zero set of h h .