zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

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