Remarks on the Yamabe problem and the Palais-Smale condition
Let M be a \(C^{\infty}\) compact Riemannian manifold of dimension \(n\geq 3\). Let g be its metric and R its scalar curvature. It is well known that the problem of finding a metric g' conformal to g such that the scalar curvature \(\bar R\) of g' is constant is reduced to minimize the functional \[ \psi (u)=\int_{M}(\gamma | \nabla | u^ 2+R(x)u^ 2)dM,\quad \gamma =4(n-1)/n-2 \] on the manifold \(V=\{u\in H^ 1(M)| \int_{M}| u|^{2*}dM=1\}\), where \(H^ 1(M)\) is the Sobolev space on M and \(2*=2n/(n-2)\). In this paper the authors prove that \(\psi /_ V\) satisfies the Palais-Smale condition in the strip ]- \(\infty,\gamma S]\) where S is the best constant for the Sobolev embedding. Moreover it is proved that the Palais-Smale condition is not satisfied at the level \(\gamma\) S for any compact \(C^{\infty}\)- Riemannian manifold. We recall that meanwhile \textit{R. Schoen} [J. Differ. Geom. 20, 479-495 (1984; Zbl 0576.53028)] has proved that, if M is ''conformally'' different from \(S^ n\), \(\mu\equiv \inf \{\psi (u):\) \(u\in V\}<\gamma S\) and consequently \(\mu\) is a minimum.
scientific article; zbMATH DE number 3986096
scientific article