Prescribing Scalar Curvature on Sn and Related Problems, Part I

Abstract

Let ( S n , g 0) be the standard n-sphere. The following question was raised by L. Nirenberg. Which function K( x) on S 2 is the Gauss curvature of a metric g on S 2 conformally equivalent to g 0? Naturally one may ask a similar question in higher dimensional case, namely, which function K( x) on S n is the scalar curvature of a metric g on S n conformally equivalent to g 0? We give some apriori estimates for solutions of the prescribed scalar curvature equations for n ≥ 3 and some existence results which are quite natural extensions of previous results of Chang and Yang ([CY2]) and Bahri and Coron ([BC2]) for n = 2, 3. For n = 3 such estimates have been obtained by Schoen and Zhang (see [Sc2] and [Z]). As a byproduct of the blow up analysis completed here for n ≥ 3 we have essentially extended all the results in [Lil-3] for n = 3, 4 to higher dimensional cases. In [Lil-3], a procedure of gluing approximate solutions into genuine solutions has been completed for subcritical equations corresponding to the scalar curvature equations. This procedure is an adaptation of some "gluing technique" for periodic ode′s and periodic subcritical pde′s via variational methods originally introduced by Séré ([Se]) and Coti-Zelati and Rabinowitz ([CR1-2]). See also an earlier related paper by Coti-Zelati et al. ([CES]). Now we can pass to limit and obtain our results for n ≥ 3. In particular we show that C ∞ scalar curvature functions are C 0 dense among functions which are positive somewhere. This extends the L p density result of Bourguignon and Ezin in [BoE]. The related critical exponent equations −Δ u = K( x) u ( n+2)/( n−2) in R n with K( x) being periodic in one of the variables are also studied and infinitely many positive solutions (module translations by its periods) are obtained under some additional mild hypotheses on K( x). The main results in this paper have been announced in [Li5].