*(English)*Zbl 1130.58010

The very interesting paper under review deals with the existence of different solutions to a nonlinear elliptic problem on Riemannian manifolds. Precisely, let $(M,g)$ be a ${C}^{\infty},$ compact and connected Riemannian manifold without boundary of dimension $n\ge 3\xb7$ Consider the problem

for $p\in (2,{2}^{*})$ with ${2}^{*}$ 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 $M\xb7$ Precisely, set $\text{cat}\left(M\right)$ for the Ljusternik-Schnirelmann category of $M$ in itself, and ${P}_{t}\left(M\right)$ for its PoincarĂ© polynomial.

The main results of the paper are as follows:

Theorem A. For small enough $\epsilon >0$ there exist at least $\text{cat}\left(M\right)+1$ non-constant distinct solutions of the problem $(*)$.

Theorem B. Assume that for small enough $\epsilon >0$ all the solutions of $(*)$ are non-degenerate. Then there are at least $2{P}_{1}\left(M\right)-1$ solutions.