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 $$ (\phi_p(x'))'+f(t,x)=0,\quad x(0)=0=x(T), \tag 1$$ with $\phi_p(u)=|u|^{p-2}u$, $1<p<\infty$, and assume $$ a(t)\leq\liminf_{|u|\to\infty}\frac{f(t,u)}{\phi_p(u)} \leq\limsup_{|u|\to\infty}\frac{f(t,u)}{\phi_p(u)}\leq b(t). \tag 2$$ The existence of solutions to (1) is obtained from conditions on the eigenvalues of the problem $$ (\phi_p(x'))'+\lambda w(t)\phi_p(x)=0,\quad x(0)=0=x(T), \tag 3$$ with $w=a$ and $w=b$. It is first proved that for positive weights $w$, the problem (3) has a sequence of eigenvalues $0<\lambda_1(w)<\ldots<\lambda_k(w)<\ldots$ which depend monotonically on the weight. The existence of a solution to $$ (\phi_p(x'))'+g(x)x'+f(t,x)=0,\quad x(0)=0=x(T), $$ is obtained assuming $$ \limsup_{|u|\to\infty}\frac{f(t,u)}{\phi_p(u)}\leq b(t), $$ where $b\in L^1(0,T)$ is positive and $\lambda_1(b)>1$. Similarly, the existence of a solution to (1) follows assuming (2), where $a$ and $b\geq a$ are positive and for some $k\geq 2$, $\lambda_{k-1}(a)<1<\lambda_k(b)$. 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
WorldCat.org
Full Text: DOI
References:
[1] Beesack, P. R.; Das, K. M.: Extensions of Opial’s inequality. Pacific J. Math. 26, 215-232 (1968) · Zbl 0162.07901
[2] Boccardo, L.; Dràbek, P.; Giachetti, D.; Kuček, M.: Generalization of Fredholm alternative for nonlinear differential operators. Nonlinear anal. 10, 1083-1103 (1986) · Zbl 0623.34031
[3] Brown, R. C.; Hinton, D. B.: Opial’s inequality and oscillation of 2nd order equations. Proc. amer. Math. soc. 125, 1123-1129 (1997) · Zbl 0866.34026
[4] Cuesta, M.; Gossez, J. -P.: A variational approach to nonresonance with respect to the fučik spectrum. Nonlinear anal. 19, 487-500 (1992) · Zbl 0768.34025
[5] Del Pino, M.; Elgueta, M.; Manásevich, R.: A homotopic deformation along p of a Leray-Schauder degree result and existence for (|u’|p-2u’)’+$f(t,u)=0, u(0)=u(T)=0$, p1. J. differential equations 80, 1-13 (1989) · Zbl 0708.34019
[6] Fonda, A.; Habets, P.: Periodic solutions of asymptotically positively differential equations. J. differential equations 81, 68-97 (1989) · Zbl 0692.34041
[7] Habets, P.; Metzen, G.: Existence of periodic solutions of Duffing equations. J. differential equations 78, 1-32 (1989) · Zbl 0676.34025
[8] Garcıá-Huidobro, M.; Manásevich, R.; Zanolin, F.: A Fredholm-like result for strongly nonlinear second order ODE’s. J. differential equations 114, 132-167 (1994) · Zbl 0835.34028
[9] Liu, B.: The stability of the equilibrium of a conservative system. J. math. Anal. appl. 202, 133-149 (1996) · Zbl 0873.34042
[10] Manásevich, R.; Mawhin, J.: Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. differential equations 145, 367-393 (1998) · Zbl 0910.34051
[11] Manásevich, R.; Zanolin, F.: Time-mappings and multiplicity of solutions for the one-dimensional p-Laplacian. Nonlinear anal. 21, 269-291 (1993) · Zbl 0792.34021
[12] Mawhin, J.: Topological degree methods in nonlinear boundary value problems. Cbms 40 (1979) · Zbl 0414.34025
[13] Talenti, G.: Best constant in Sobolev inequality. Ann. mat. Pura appl. 110, 353-372 (1976) · Zbl 0353.46018
[14] Zhang, M.: Nonuniform nonresonance at the first eigenvalue of the p-Laplacian. Nonlinear anal. 29, 41-51 (1997) · Zbl 0876.35039
[15] Zhang, M.: Nonresonance conditions for asymptotically positively homogeneous differential systems: the fučik spectrum and its generalization. J. differential equations 145, 332-366 (1998) · Zbl 0913.34051
[16] Zhang, M.: A relationship between the periodic and the Dirichlet BVPs of singular differential equations. Proc. roy. Soc. Edinburgh sect. A 128, 1099-1114 (1998) · Zbl 0918.34025