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)
An integral system and the Lane-Emden conjecture. (English) Zbl 1176.35067

Summary: We consider the system of integral equations in n :

u(x)= n 1 |x-y| n-μ v q (y)dy,v(x)= n 1 |x-y| n-μ u p (y)dy,

with 0<μ<n. Under some integrability conditions, we obtain radial symmetry of positive solutions by using the method of moving planes in integral forms. In the special case when μ=2, we show that the integral system is equivalent to the elliptic PDE system

-Δ[u=v q (x)],-Δ[v=u p (x)]

in n . Our symmetry result, together with non-existence of radial solutions by E. Mitidieri [Commun. Partial Differ. Equations 18, No. 1–2, 125–151 (1993; Zbl 0816.35027], implies that, under our integrability conditions, the PDE system possesses no positive solution in the subcritical case. This partially solved the well-known Lane-Emden conjecture.

MSC:
35J60Nonlinear elliptic equations
45G15Systems of nonlinear integral equations
35J46First-order elliptic systems
35J47Second-order elliptic systems
35J48Higher-order elliptic systems