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)
Locating the peaks of least-energy solutions to a semilinear Neumann problem. (English) Zbl 0796.35056

We continue our study initiated in [C.-S. Lin, W.-M. Ni and I. Takagi, J. Differ. Equations 72, No. 1, 1-27 (1988; Zbl 0676.35030)] and [(*) W.-M. Ni and I. Takagi, Commun. Pure Appl. Math. 44, No. 7, 819-851 (1991; Zbl 0754.35042)] on the shape of certain solutions to a semilinear Neumann problem arising in mathematical models of biological pattern formation. Let Ω be a bounded domain in N with smooth boundary Ω and let ν be the unit outer normal to Ω. We consider the Neumann problem for certain semilinear elliptic equations including

dΔu-u+u p =0andu>0inΩ,u/ν=0onΩ,( BVP ) d

where d>0 and p>1 are constants. This problem is encountered in the study of steady-state solutions to some reaction-diffusion systems in chemotaxis as well as in morphogenesis.

Assume that p is subcritical, i.e., 1<p<(N+2)/(N-2) when N3 and 1<p<+ when N=2. Then we can apply the mountain-pass lemma to obtain a least-energy solution u d to (BVP) d , by which it is meant that u d has the smallest energy J d (u)=1 2 Ω (d|u| 2 +u 2 )dx-(p+1) -1 Ω u + p+1 dx, where u + =max{u,0}, among all the solutions to (BVP) d . It turns out that u d 1 if d is sufficiently large, whereas u d exhibits a “point- condensation phenomenon” as d0. More precisely, when d is sufficiently small, u d has only one local maximum over Ω ¯ (thus it is the global maximum), and the maximum is achieved at exactly one point P d on the boundary. Moreover, u d (x)0 as d0 for all xΩ, while maxu d 1 for all d>0. Hence, a natural question raised immediately is to ask where on the boundary the maximum point P d is situated, and it is the purpose of the present paper to answer this question. Indeed, we show that H(P d ), the mean curvature of Ω at P d , approaches the maximum of H(P) over Ω as d0, as was announced in (*).


MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
35J20Second order elliptic equations, variational methods