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)
Non-constant steady-state solutions for Brusselator type systems. (English) Zbl 1156.35022

Summary: We are concerned with the following stationary system:

-θΔu=λ(1-(b+1)u+bu m v)inΩ,-Δv=λa 2 (u-u m v)inΩ,

subject to homogeneous Neumann boundary conditions. Here Ω N (N1) is a smooth and bounded domain and a,b,m,λ,θ are positive parameters. The particular case m=2 corresponds to the steady-state Brusselator system. We establish existence and non-existence results for non-constant positive classical solutions. In particular, we provide upper and lower bounds for solutions which allows us to extend the previous works in the literature without any restriction on the dimension N1. Our analysis also emphasizes the role played by the nonlinearity u m . The proofs rely essentially on various types of a priori estimates.

MSC:
35J55Systems of elliptic equations, boundary value problems (MSC2000)
35J60Nonlinear elliptic equations
35B45A priori estimates for solutions of PDE