Infinitely many solutions for the prescribed scalar curvature problem on Sᴺ

Abstract

In this paper, we consider the simplest case, i.e., K is rotationally symmetric, K = K(r), r = |y|. It follows from the Pohozaev identity (1.2) that (1.6) has no solution if K'(r) has fixed sign. Thus we assume that K is positive and not monotone. On the other hand, Bianchi [5] showed that any solution of (1.6) is radially symmetric if there is an rₒ > 0, such that K(r) is non-increasing in (0, rₒ], and non-decreasing in [rₒ,+∞). Moreover, in [6], it was proved that (1.6) has no solutions for some function K(r), which is non-increasing in (0, 1], and nondecreasing in [1,+∞). Therefore, we see that to obtain a solution for (1.6), it is natural to assume that K(r) has a local maximum at rₒ > 0. The purpose of this paper is to answer the following two questions: Q1. Does the existence of a local maximum of K guarantee the existence of a solution to (1.6)? Q2. Are there non-radially symmetric solutions to (1.6)?