The very interesting paper under review deals with the existence of different solutions to a nonlinear elliptic problem on Riemannian manifolds. Precisely, let be a compact and connected Riemannian manifold without boundary of dimension Consider the problem
for with being the critical exponent for the Sobolev immersion. The authors study the relation between the number of solutions to and the topology of the manifold Precisely, set for the Ljusternik-Schnirelmann category of in itself, and for its Poincaré polynomial.
The main results of the paper are as follows:
Theorem A. For small enough there exist at least non-constant distinct solutions of the problem .
Theorem B. Assume that for small enough all the solutions of are non-degenerate. Then there are at least solutions.