Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]