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 interior spike solutions for some singular perturbation problems. (English) Zbl 0944.35021
The paper deals with positive solutions to singularly perturbed semilinear elliptic problems of the following form: $$ - \varepsilon^2\Delta u=f(u) \quad \text{in } \Omega u=0 \quad \text{ or} \quad \frac{\partial u}{\partial \nu}=0 \quad\text{on }\partial \Omega .$$ Here $ \Omega \subseteq \bbfR^n $ is a bounded domain with smooth boundary $\partial \Omega$ and $f:\bbfR \rightarrow \bbfR$ is smooth; a typical choice is $f(u)=-u+u^p (p>1)$. The purpose of the paper is to investigate the existence of single interior spike solutions. A family of solutions $\{u_\varepsilon\}$ is called single peaked if: $(i)$ the energy is bounded for any $\varepsilon>0$; $(ii)$ $u_\varepsilon$ has only one local maximum point $P_\varepsilon$, $P_\varepsilon \rightarrow P_0$, $u_\varepsilon \rightarrow 0$ in $C^1_{\text{loc}}(\overline \Omega \setminus P_0)$ and $u_\varepsilon(P_\varepsilon) \rightarrow \alpha>0$ as $\varepsilon \rightarrow 0$. The limiting point $P_0$ is called a boundary spike if $P_0 \in \partial \Omega$, respectively an interior spike if $P_0 \in \Omega$. In both cases the problem of existence and location of spikes has stimulated quite a few papers in recent years. The paper provides a unified approach to this problem under rather general assumptions on the function $f$ (both for Dirichlet and for Neumann homogeneous boundary conditions); in particular, necessary and sufficient conditions for the existence of interior spike solutions are given. A variety of methods is used to this purpose, including in particular weak convergence of measures and Lyapunov-Schmidt reduction.

35J55Systems of elliptic equations, boundary value problems (MSC2000)
35B25Singular perturbations (PDE)
35B65Smoothness and regularity of solutions of PDE
Full Text: DOI