The scalar curvature problem on \(S^n\): An approach via Morse theory (Q1614899)

In this paper the author proves the existence of positive solutions for the equation \[ -4{n-1 \over n-2 }\Delta u +n(n-1)u = R' u^{{n+2 \over n-2 }} \] on the sphere \(S^n\) (\(n \geq 3\)), where \(\Delta\) denotes the Laplace-Beltrami operator with respect to the standard metric on \(S^n\) and \(R'\) is a small perturbation of a constant. Applying perturbation methods and placing suitable non-degeneracy conditions on the perturbations the author reduces the problem from an infinite-dimensional problem to a finite-dimensional one in order to obtain the existence.