Nonlinear elliptic equations with critical Sobolev exponent on compact Riemannian manifolds (Q1300119)

Let \(M\) be a compact Riemannian manifold of dimension \(n\geq 3\), let \(a\), \(f\), and \(h\) be smooth functions on \(M\), let \(1 < q < {n+2 \over n-2}\). Generalizing the classical Yamabe problem and the problem of prescribed scalar curvature the author looks for positive smooth solutions \(u\) of \[ \Delta u + au = fu^{n+2 \over n-2} + hu^q . \] It is assumed that the operator \(L = \Delta + a\) is coercive. Existence of solutions is proved under various additional assumptions on \(f\) and \(h\). The mountain pass lemma is used as an essential tool for the proof. As for uniqueness results it is shown that if \(a\), \(f\), and \(h\) are constant and positive and Ricci curvature satisfies \[ \operatorname {Ric} \geq {4(n-1) \over n(n-2)} ag \] then there is a unique solution which is a constant. This uses a Bochner-Łichnerowicz-Weitzenböck formula.