Prescribing scalar curvature on \(S^ N\). I: A priori estimates. (English) Zbl 1043.53028

Chen, Chiun-Chuan; Lin, Chang-Shou

J. Differ. Geom. 57, No. 1, 67-171 (2001).

In this paper, the authors analyze in great detail the bubbling behavior for a sequence of solutions \(w_i\) of \[ Lw_i+R_iw_i^{(n+2)/(n-2)}=0 \] on \({\mathbb S}^n,\) where \(L\) is the conformal Laplacian of \(({\mathbb S}^n,g_0)\), and \(R_i=n(n-2)+t_i\hat{R},\) \(\hat{R}\in C^1({\mathbb S}^n).\) As \(t_i\to 0+,\) they prove, among other things, the location of blow-up points, the spherical Harnack inequality near each blow-up point and the asymptotic formulas for the interaction of different blow-up points. This is the first step for computing the degree-counting formula.

Reviewer: Alberto Parmeggiani (Bologna)

MSC:

53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
35B45	A priori estimates in context of PDEs
35J60	Nonlinear elliptic equations
58J60	Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)

Keywords:

scalar curvature; conformal Laplacian; bubbling behavior