Advanced Search

Query:

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

Format:

Display: entries per page entries

Zbl 0822.35046
Benci, Vieri; Cerami, Giovanna
Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. (English)
[J] Calc. Var. Partial Differ. Equ. 2, No.1, 29-48 (1994). ISSN 0944-2669; ISSN 1432-0835/e

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.
[M.Degiovanni (Brescia)]

MSC 2000:: *35J65 (Nonlinear) BVP for (non)linear elliptic equations
35J20 Second order elliptic equations, variational methods
58E05 Abstract critical point theory

Keywords: multiple positive solutions; semilinear elliptic equations; Morse index; Ljusternik-Schnirelman category

Cited in: Zbl 1199.35082

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]