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)
Nonuniform nonresonance of semilinear differential equations. (English) Zbl 0962.34062

Consider the Dirichlet problem

(φ p (x ' )) ' +f(t,x)=0,x(0)=0=x(T),(1)

with φ p (u)=|u| p-2 u, 1<p<, and assume

a(t)lim inf |u| f(t,u) φ p (u)lim sup |u| f(t,u) φ p (u)b(t)·(2)

The existence of solutions to (1) is obtained from conditions on the eigenvalues of the problem

(φ p (x ' )) ' +λw(t)φ p (x)=0,x(0)=0=x(T),(3)

with w=a and w=b. It is first proved that for positive weights w, the problem (3) has a sequence of eigenvalues 0<λ 1 (w)<...<λ k (w)<... which depend monotonically on the weight. The existence of a solution to

(φ p (x ' )) ' +g(x)x ' +f(t,x)=0,x(0)=0=x(T),

is obtained assuming

lim sup |u| f(t,u) φ p (u)b(t),

where bL 1 (0,T) is positive and λ 1 (b)>1. Similarly, the existence of a solution to (1) follows assuming (2), where a and ba are positive and for some k2, λ k-1 (a)<1<λ k (b). These results are based on degree arguments. In a last section, best Sobolev constants are obtained which imply estimates on the first eigenvalue of (3). These are used in examples.

MSC:
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
34B15Nonlinear boundary value problems for ODE
34L30Nonlinear ordinary differential operators