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Zbl 1130.58010
Benci, Vieri; Bonanno, Claudio; Micheletti, Anna Maria
On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds.
(English)
[J] J. Funct. Anal. 252, No. 2, 464-489 (2007). ISSN 0022-1236

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 $$\cases -\varepsilon^2\Delta_gu+u-u\vert u\vert ^{p-2}=0,\\ 0<u\in H^1_g(M) \endcases \tag *$$ 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.
[Dian K. Palagachev (Bari)]
MSC 2000:
*58J05 Elliptic equations on manifolds, general theory
35J60 Nonlinear elliptic equations

Keywords: nonlinear elliptic equation; Riemannian manifold; Lyusternik-Shnirelmann category; bump solutions

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