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)
Resonance problems for the p-Laplacian. (English) Zbl 0940.35087

Using variational arguments the authors prove the existence of a weak solution for the boundary value problem

-Δ p u-λ|u| p-2 u+f(x,u)=0inΩ,u=0onΩ,

where Δ p u=div(|Du| p-2 Du), p>1, Ω is a bounded domain of N , λ and f:Ω× is a bounded Carathéodory function such that there exist lim t± f(x,t)=f ± (x) a.e. in Ω, with either

v>0 f + v+ v<0 f - v>0,or v>0 f + v+ v<0 f - v<0or(( LL ) λ + ,( LL ) λ - )

for all vKer(-Δ p -λ){0}, and also there is gL p/(p-1) (Ω) such that


The conditions (LL) λ ± are the standard Landesman – Laser conditions for resonance problems, and of course are trivially satisfied whenever λ is not an eigenvalue.

35J65Nonlinear boundary value problems for linear elliptic equations
35P30Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory
35A15Variational methods (PDE)
35A05General existence and uniqueness theorems (PDE) (MSC2000)