zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. (English) Zbl 0822.35046
The authors study the number of solutions $u$ of the problem $$- \varepsilon \Delta u+u= f(u),\ \ u>0 \quad \text{in } \Omega, \qquad u=0 \quad \text{on }\partial \Omega, \tag P$\sb \varepsilon$ $$ where $\varepsilon>0$ and $f: \bbfR\sp +\to \bbfR$ is subcritical and superlinear at 0 and at infinity. They show that there exists $\varepsilon\sp*>0$ such that, if $0< \varepsilon\leq \varepsilon\sp*$ and all solutions of $(\text{P}\sb \varepsilon)$ are non-degenerate, we have $$\sum\sb{u\in {\cal K}} t\sp{\mu (u)}= t{\cal P}\sb t (\Omega)+ t\sp 2 [{\cal P}\sb t (\Omega)-1]+ t(1+ t){\cal Q} (t),$$ where ${\cal K}$ is the set of solutions of $(\text{P}\sb \varepsilon)$, $\mu(u)$ is the Morse index of $u$, ${\cal P}\sb t (\Omega)$ is the Poincaré polynomial of $\Omega$ and ${\cal Q}$ is a suitable polynomial with non- negative integer coefficients. Actually, this is a consequence of a more general statement, where ${\cal K}$ is required only to be discrete. Also an estimate of the cardinality of ${\cal K}$ in terms of the Ljusternik- Schnirelman category of $\Omega$ is given.

35J65Nonlinear boundary value problems for linear elliptic equations
35J20Second order elliptic equations, variational methods
58E05Abstract critical point theory
Full Text: DOI
[1] Benci, V.: A new approach to the Morse-Conley theory and applications. Ann. Mat. Pura Appl.158, 231-305 (1991) · Zbl 0778.58011 · doi:10.1007/BF01759307
[2] Benci, V.: Introduction to Morse theory. A new approach, (to appear) · Zbl 0823.58008
[3] Benci, V., Cerami, G.: The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Ration. Mech. Anal.114, 79-93 (1991) · Zbl 0727.35055 · doi:10.1007/BF00375686
[4] Benci, V., Cerami, G., Passaseo, D.: On the number of the positive solutions of some nonlinear elliptic problems. In: Nonlinear analisys. A tribute in honour of G. Prodi. Quaderno Scuola Normale Superiore, Pisa, 1991, pp. 93-109 · Zbl 0838.35040
[5] Benci, V., Giannoni, F.: In preparation.
[6] Candela, A.M.: Remarks on the number of solutions for a class of nonlinear elliptic problems. Preprint. · Zbl 0784.35031
[7] Cerami, G., Passaseo, D.: Existence and multiplicity of positive solutions for nonlinear elliptic problems in exterior domains with rich topology. J. Nonlinear Anal. T.M.A.18, 103-119 (1992). · Zbl 0810.35024
[8] Chang, K.C.: Infinite dimensional Morse theory and its applications. Semin. Mat. Sup?r.97, Press of the University of Montreal (1986)
[9] Dold, A.; Lectures on algebraic topology. Berllin, Heidelberg, New York: Springer 1972 · Zbl 0234.55001
[10] Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in ?N. Mathematical analysis and applications. Part A: Advances in mathematics supplementary studies, vol. 7A, pp. 369-402 (1980)
[11] Lions, P.L.: The concentration ? compactness principle in the calculus of variations. The locally compact case. I-II. Ann. Inst. Henri Poincar?, Anal. Non Lineaire1, 223-283, 109-145 (1984) · Zbl 0704.49004
[12] Marino, A., Prodi, G.: Metodi perturbativi nella teoria di Morse. Boll. U.M.I.11 (4), 1-32 (1975) · Zbl 0311.58006
[13] Strauss, W.: Existence of solitary waves in higher dimensions. Commun. Math. Phys.55, 149-162 (1977) · Zbl 0356.35028 · doi:10.1007/BF01626517