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)
Integral averages and oscillation criteria for half-linear partial differential equation. (English) Zbl 1046.35034
The aim of this paper is to extend the technique of weighted integral averages to the half-linear partial differential equation $$\Delta_p u+ c(x)\Phi(u)= 0,\tag E$$ where $\Delta_p u\equiv \text{div}(\Vert u\Vert^{p-2}\nabla u)$, $p> 1$ is the $p$-Laplacian, $\Phi(u)=\vert u\vert^{p-2} u=\vert u\vert^{p-1}\text{sgn }u$, and $x= (x_i)^n_{i=1}\in \bbfR^n$. On the one hand, the author obtains new oscillation criteria for (E) which can remove the disadvantage of some previous theorems, and on the other hand he shows that this technique allows to obtain oscillation criteria not only for the exterior of a ball, but also for different types of unbounded domains. Some illustrative examples are given.

35J60Nonlinear elliptic equations
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
Full Text: DOI
[1] Atakarryev, M.; Toraev, A.: Oscillation and non-oscillation criteria of knezer type for elliptic nondivergent equations in unlimited areas. Proc. acad. Sci. turkmen. SSR 6, 3-10 (1986) · Zbl 0674.35022
[2] Dı\acute{}az, J. I.: Nonlinear partial differential equations and free boundaries. Elliptic equations (1985)
[3] Došłý, O.: Methods of oscillation theory of half-linear second order differential equations. Czech math. J. 125, No. 2, 657-671 (2000) · Zbl 1079.34512
[4] Došlý, O.: Oscillation criteria for half-linear second order differential equations. Hiroshima math. J. 28, 507-521 (1998) · Zbl 0920.34042
[5] O. Došlý, A. Lomtatidze, Oscillation and nonoscillation criteria for half-linear second order differential equations, submitted for publication
[6] Došlý, O.; Mařı\acute{}k, R.: Nonexistence of positive solutions for PDE’s with p-Laplacian. Acta math. Hungar. 90, No. 1--2, 89-107 (2001)
[7] Fiedler, F.: Oscillation criteria of Nehari-type for Sturm--Liouville operators and elliptic operators of second order and the lower spectrum. Proc. roy. Soc. edinb. 109A, 127-144 (1988) · Zbl 0657.34036
[8] Jaroš, J.; Kusano, T.; Yoshida, N.: A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order. Nonlin. anal. TMA 40, 381-395 (2000) · Zbl 0954.35018
[9] Kandelaki, N.; Lomtatidze, A.; Ugulava, D.: On the oscillation and nonoscillation of a second order half-linear equation. Georgian math. J. 7, No. 2, 347-353 (2000)
[10] Kong, Q.: Interval criteria for oscillation of second-order linear ordinary differential equations. J. math. Anal. appl. 229, 258-270 (1999) · Zbl 0924.34026
[11] Kusano, T.; Naito, Y.: Oscillation and nonoscillation criteria for second order quasilinear differential equations. Acta math. Hungar. 76, 81-99 (1997) · Zbl 0906.34024
[12] Kusano, T.; Naito, Y.; Ogata, A.: Strong oscillation and nonoscillation of quasilinear differential equations of second order. Diff. equations dyn. Syst. 2, 1-10 (1994) · Zbl 0869.34031
[13] Li, H. J.: Oscillation criteria for half-linear second order differential equations. Hiroshima math. J. 25, 571-583 (1995) · Zbl 0872.34018
[14] R. Mařı\acute{}k, Hartman--Wintner type theorem for PDE with p-Laplacian, EJQTDE 18 (2000), Proc. 6th Coll. QTDE, pp. 1--7
[15] Mařı\acute{}k, R.: Oscillation criteria for PDE with p-Laplacian via the Riccati technique. J. math. Anal. appl. 248, 290-308 (2000) · Zbl 0964.35044
[16] Moss, W. F.; Piepenbrick, J.: Positive solutions of elliptic equations. Pacific J. Math. 75, No. 1, 219-226 (1978) · Zbl 0381.35026
[17] Müller-Pfeiffer, E.: Oscillation criteria of Nehari-type for the schödinger equation. Math. nachr. 96, 185-194 (1980) · Zbl 0454.35007
[18] Naito, M.; Naito, Y.; Usami, H.: Oscillation theory for semilinear ellitpic equations with arbitrary nonlinearities. Funkcial. evkac. 40, 41-553 (1997) · Zbl 0883.35008
[19] Noussair, E. W.; Swanson, C. A.: Positive solutions of semilinear Schrödinger equations in exterior domains. Ind. uni. Math. J. 6, 993-1003 (1979) · Zbl 0397.35014
[20] Noussair, E. W.; Swanson, C. A.: Oscillation of semilinear elliptic inequalities by Riccati equation. Can. math. J. 22, No. 4, 908-923 (1980) · Zbl 0395.35027
[21] Philos, Ch.G.: Oscillation theorems for linear differential equations of second order. Arch. math (Basel) 53, 483-492 (1989) · Zbl 0661.34030
[22] Schminke, U. -W.: The lower spectrum of Schrödinger operators. Arch. rational mech. Anal. 75, 147-155 (1989)
[23] Toraev, A.: Ob oscilacii resenij ellipticeskich uravnenij. Dokl. akad. Nauk SSSR 280, 300-303 (1985)
[24] Wang, Q. R.: Oscillation and asymptotics for second-order half-linear differential equations. Appl. math. Comp. 122, 253-266 (2001) · Zbl 1030.34031