The scalar curvature problem on $S^n$: an approach via Morse theory

Abstract.

We prove the existence of positive solutions for the equation on $S^n - 4 \times \frac{(n-1)}{(n-2)}\Delta_{g_0} u + n (n-1) u = (1+\varepsilon K_0(x))u^{2^*-1}$ , where $\Delta_{g_0}$ is the Laplace-Beltrami operator on $S^n, 2^*$ is the critical Sobolev exponent, and $\varepsilon$ is a small parameter. The problem can be reduced to a finite dimensional study which is performed via Morse theory.