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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 , compact and connected Riemannian manifold without boundary of dimension n3· Consider the problem

-ε 2 Δ g u+u-u|u| p-2 =0,0<uH g 1 (M)(*)

for p(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 cat(M) for the Ljusternik-Schnirelmann category of M in itself, and P t (M) for its PoincarĂ© polynomial.

The main results of the paper are as follows:

Theorem A. For small enough ε>0 there exist at least cat(M)+1 non-constant distinct solutions of the problem (*).

Theorem B. Assume that for small enough ε>0 all the solutions of (*) are non-degenerate. Then there are at least 2P 1 (M)-1 solutions.

58J05Elliptic equations on manifolds, general theory
35J60Nonlinear elliptic equations