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)
On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. (English) Zbl 0838.35009
The authors consider problem (1): $\varepsilon^2\Delta u- u+ f(u)= 0$ and $u> 0$ in $\Omega$, $u= 0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\bbfR^n$, with smooth boundary $\partial\Omega$, and $f$ is a suitable function $\bbfR\to \bbfR$; the particular case $f(t)= t^p$, $1< p< (n+ 2)/(n- 2)$ is allowed. They state that, as $\varepsilon\to 0$, a least energy solution $u_\varepsilon$ to (1) has at most one local maximum which is achieved at exactly one point $P_\varepsilon\in \Omega$; furthermore $u_\varepsilon\to 0$ except at $P_\varepsilon$ and $d(P_\varepsilon, \partial\Omega)\to \max_{P\in \Omega} d(P, \partial\Omega)$, where $d$ denotes the distance function. Their approach is based on an asymptotic formula for the least positive critical value $c_\varepsilon$ of the energy $J_\varepsilon$ (i.e. $J_\varepsilon(u_\varepsilon)= c_\varepsilon$). In particular, they show that the dominating correction term in the expansion for $c_\varepsilon$, involves $d(P_\varepsilon, \partial\Omega)$ and is of order $\exp(- 1/\varepsilon)$. They make use of the vanishing viscosity method and methods developed earlier for the corresponding Neumann problem [the first author and {\it I. Takagi}, Commun. Pure Appl. Math. 44, No. 7, 819-851 (1991; Zbl 0754.35042) and Duke Math. J. 70, No. 2, 247-281 (1993; Zbl 0796.35056)].
Reviewer: D.Huet (Nancy)

35B25Singular perturbations (PDE)
35J60Nonlinear elliptic equations
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
Full Text: DOI