The scalar curvature problem on Sⁿ: an approach via Morse theory

We prove the existence of positive solutions for the equation on Sn −4 ×(n−1)/(n−2)∆g0 u + n(n − 1)u = (1 + εK0(x))u2∗−1 , where ∆g0 is the Laplace-Beltrami operator on Sn, 2∗ is the critical Sobolev exponent, and ε is a small parameter. The problem can be reduced to a finite dimensional study which is performed via Morse theory