×

A nonoscillation theorem for superlinear Emden-Fowler equations with near-critical coefficients. (English) Zbl 1125.34023

The authors deal with the oscillatory behavior of solutions of the superlinear Emden-Fowler differential equation
\[ y''(x)+a(x)| y(x)|^{\gamma-1}y(x)=0,\quad x>0,\tag{1} \]
where \(\gamma>1\) and \(a(x)\) is a positive continuous function on \((0,\infty)\). One of the known results says that if \(a(x)= x^{-(\gamma+3)/2}\log^{-\sigma}(x)\), where \(\sigma>0\), then all solutions of (1) are nonoscillatory. In this paper, this result is extended to include a class of coefficients in which the above condition with \(\log(x)\) can be replaced by \(\log\log(x)\), or \(\log\log\log(x)\) and so on.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Atkinson, F. V.; Peletier, L. A., Ground states of \(- \Delta u = f(u)\) and the Emden-Fowler equation, Arch. Ration. Mech. Anal., 93, 103-127 (1986) · Zbl 0606.35029
[2] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36, 437-477 (1983) · Zbl 0541.35029
[3] Chiou, K. L., A second order nonlinear oscillation theorem, SIAM J. Appl. Math., 21, 221-224 (1971) · Zbl 0231.34029
[4] Chiou, K. L., A nonoscillation theorem for the superlinear case of second order differential equations \(y'' + y F(y^2, x) = 0\), SIAM J. Appl. Math., 23, 456-459 (1972) · Zbl 0229.34035
[5] Coffman, C. V., Uniqueness of the ground state solution for \(\Delta u - u + u^3 = 0\) and a variational characterization of other solutions, Arch. Ration. Mech. Anal., 46, 81-95 (1972) · Zbl 0249.35029
[6] Coffman, C. V.; Wong, James S. W., On a second-order nonlinear oscillation problem, Trans. Amer. Math. Soc., 147, 357-366 (1970) · Zbl 0223.34030
[7] Coffman, C. V.; Wong, James S. W., Oscillation and nonoscillation of solutions of generalized Emden-Fowler equations, Trans. Amer. Math. Soc., 167, 399-434 (1972) · Zbl 0278.34026
[8] Erbe, L. H.; Muldowney, J. S., On the existence of oscillatory solutions to nonlinear differential equations, Anali di Mat. Pure Appl., 109, 23-38 (1976) · Zbl 0345.34022
[9] Erbe, L. H.; Muldowney, J. S., Nonoscillation results for second order nonlinear differential equations, Rocky Mountain Math. J., 12, 635-642 (1982) · Zbl 0516.34030
[10] Erbe, L. H.; Tang, M., Uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball, J. Differential Equations, 138, 351-379 (1997) · Zbl 0884.34025
[11] Heidel, J. W., Uniqueness, continuation and nonoscillation for a second-order nonlinear differential equation, Pacific J. Math., 32, 715-721 (1970) · Zbl 0188.14301
[12] Kaper, H.; Kwong, Man Kam, A non-oscillation theorem for the Emden-Fowler equation: Ground states for semilinear elliptic equations with critical exponents, J. Differential Equations, 75, 158-185 (1988) · Zbl 0669.35035
[13] Kiguradze, I. T., On oscillatory conditions of the equation \(u'' + a(t) | u |^\gamma sgn u = 0\), Casopis Pest. Mat., 87, 492-495 (1962), (in Russian) · Zbl 0138.33504
[14] Kiguradze, I. T., On the oscillatory and monotone solutions of ordinary differential equations, Arch. Math. Scripta Fac. Sci. Nat., 14, 21-44 (1978) · Zbl 0408.34033
[15] Kiguradze, I. T.; Chanturia, T. A., Asymptotic Properties of Solutions of Non-Autonomous Ordinary Differential Equations (1993), Kluwer: Kluwer Dordrecht · Zbl 0782.34002
[16] Kurzweil, J., A note on oscillatory solutions of the equation \(y'' + f(x) y^{2 n - 1} = 0\), Casopis Pest. Mat., 85, 357-358 (1960), (in Russian) · Zbl 0129.06204
[17] Kwong, Man Kam, Uniqueness of positive solutions of \(\Delta u - u + u^p = 0\) in \(R^n\), Arch. Ration. Mech. Anal., 105, 243-266 (1989) · Zbl 0676.35032
[18] Kwong, Man Kam; Li, Yi, Uniqueness of radial solutions of semilinear elliptic equations, Trans. Amer. Math. Soc., 333, 339-363 (1992) · Zbl 0785.35038
[19] Kwong, Man Kam; Wong, James S. W., A nonoscillation theorem for sublinear Emden-Fowler equations, Nonlinear Anal., 64, 1641-1646 (2006) · Zbl 1099.34033
[20] Man Kam Kwong, James S.W. Wong, On the nonoscillation of sublinear Emden-Fowler equations, Differential Equations Dynam. Systems, in press; Man Kam Kwong, James S.W. Wong, On the nonoscillation of sublinear Emden-Fowler equations, Differential Equations Dynam. Systems, in press · Zbl 1099.34033
[21] Kwong, Man Kam; Wong, James S. W., Second-order nonlinear oscillations—A case history, Can. Appl. Math. Q., 14, 199-208 (2006) · Zbl 1147.34024
[22] Nehari, Z., A nonlinear oscillation problem, J. Differential Equations, 5, 452-460 (1969) · Zbl 0181.09702
[23] Nehari, Z., A nonlinear oscillation theorem, Duke Math. J., 42, 183-189 (1975) · Zbl 0385.34011
[24] Z. Nehari, Mathematical Review No. 11661, 48 (1974); Z. Nehari, Mathematical Review No. 11661, 48 (1974)
[25] Ou, C. H.; Wong, James S. W., On existence of oscillatory solutions of second-order Emden-Fowler equations, J. Math. Anal. Appl., 277, 670-680 (2003) · Zbl 1027.34039
[26] Peletier, L. A.; Serrin, James, Uniqueness of nonnegative solutions of semi-linear equations in \(R^n\), J. Differential Equations, 61, 380-397 (1986) · Zbl 0577.35035
[27] Wong, James S. W., On the generalized Emden-Fowler equation, SIAM Rev., 17, 339-360 (1975) · Zbl 0295.34026
[28] Wong, James S. W., Nonoscillation theorems for second-order nonlinear differential equations, Proc. Amer. Math. Soc., 127, 1387-1395 (1999) · Zbl 0917.34025
[29] Wong, James S. W., A nonoscillation theorem for Emden-Fowler equations, J. Math. Anal. Appl., 274, 746-754 (2002) · Zbl 1036.34039
[30] Wong, James S. W., A nonoscillation theorem for sublinear Emden-Fowler equations, Anal. Appl. (Singap.), 1, 71-79 (2003) · Zbl 1051.34028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.