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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 \(\Delta_p u\equiv \text{div}(\| u\|^{p-2}\nabla u)\), \(p> 1\) is the \(p\)-Laplacian, \(\Phi(u)=| u|^{p-2} u=| u|^{p-1}\text{sgn\,}u\), and \(x= (x_i)^n_{i=1}\in \mathbb{R}^n\).
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.

MSC:

35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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[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), (in Russian) · Zbl 0674.35022
[2] Dı́az, J. I., Nonlinear Partial Differential Equations and Free Boundaries, (Elliptic Equations, vol. I (1985), Pitman Publ: Pitman Publ London) · Zbl 0595.35100
[3] Došłý, O., Methods of oscillation theory of half-linear second order differential equations, Czech. Math. J., 125, 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; O. Došlý, A. Lomtatidze, Oscillation and nonoscillation criteria for half-linear second order differential equations, submitted for publication
[6] Došlý, O.; Mařı́k, R., Nonexistence of positive solutions for PDE’s with \(p\)-Laplacian, Acta Math. Hungar., 90, 1-2, 89-107 (2001) · Zbl 1062.35022
[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, 2, 347-353 (2000) · Zbl 0957.34032
[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. Mařı́k, Hartman-Wintner type theorem for PDE with \(p\); R. Mařı́k, Hartman-Wintner type theorem for PDE with \(p\)
[15] Mařı́k, 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, 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, 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
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