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On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds. (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 \(n\geq 3.\) Consider the problem
\[ \begin{cases} -\varepsilon^2\Delta_gu+u-u| u| ^{p-2}=0,\\ 0<u\in H^1_g(M) \end{cases} \tag \(*\) \] 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 \(M.\) Precisely, set \(\text{cat}(M)\) for the Ljusternik-Schnirelmann category of \(M\) in itself, and \(P_t(M)\) for its Poincaré polynomial.
The main results of the paper are as follows:
Theorem A. For small enough \(\varepsilon>0\) there exist at least \(\text{cat}(M)+1\) non-constant distinct solutions of the problem \((*)\).
Theorem B. Assume that for small enough \(\varepsilon>0\) all the solutions of \((*)\) are non-degenerate. Then there are at least \(2P_1(M)-1\) solutions.

MSC:

58J05 Elliptic equations on manifolds, general theory
35J60 Nonlinear elliptic equations
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