×

Sharp conditions for the existence of sign-changing solutions to equations involving the one-dimensional \(p\)-Laplacian. (English) Zbl 1157.34010

Summary: We consider the boundary value problem
\[ (|u'|^{p-2}u')'+a(x)f(u)=0,\quad 0<x<1,\quad u(0)=u(1)=0, \]
where \(p>1\). We establish sharp conditions for the existence of solutions with prescribed numbers of zeros in terms of the ratio \(f(s)/s^{p-1}\) at infinity and zero. Our argument is based on the shooting method together with the qualitative theory for half-linear differential equations.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agarwal, R. P.; Lü, H.; O’Regan, D., Eigenvalues and the one-dimensional \(p\)-Laplacian, J. Math. Anal. Appl., 266, 383-400 (2002) · Zbl 1002.34019
[2] del Pino, M.; Elgueta, M.; Manasevich, R., A homotopic deformation along \(p\) of a Leray-Schauder degree result and existence for \((| u^\prime |^{p - 2} u^\prime)^\prime + f(t, u) = 0, u(0) = u(T) = 0, p > 1\), J. Differential Equations, 80, 1-13 (1989) · Zbl 0708.34019
[3] Došlý, O., (Half-linear differential equations. Half-linear differential equations, Handbook of Differential Equations, Volume 1: Ordinary Differential Equations, vol. 1 (2004), Elsevier Science B.V.: Elsevier Science B.V. North-Holland, Amsterdam), 161-358 · Zbl 1090.34027
[4] Došlý, O.; Rehak, P., (Half-linear differential equations. Half-linear differential equations, North-Holland Mathematics Studies, vol. 202 (2005), Elsevier Science B.V.: Elsevier Science B.V. Amsterdam) · Zbl 1090.34001
[5] Elbert, Á., A half-linear second order differential equation, Qualitative theory of differential equations, Colloq. Math. Soc. János Bolyai., 30, 153-180 (1979)
[6] Garcia-Huidobro, M.; Ubilla, P., Multiplicity of solutions for a class of nonlinear second-order equations, Nonlinear Anal., 28, 1509-1520 (1997) · Zbl 0874.34021
[7] Hartman, P., Ordinary Differential Equations (1982), Birkhäuser: Birkhäuser Boston · Zbl 0125.32102
[8] He, X.; Ge, W., Twin positive solutions for the one-dimensional \(p\)-Laplacian boundary value problems, Nonlinear Anal., 56, 975-984 (2004) · Zbl 1061.34013
[9] Jaroš, J.; Kusano, T., A Picone type identity for second order half-linear differential equations, Acta Math. Univ. Comenian., 68, 137-151 (1999) · Zbl 0926.34023
[10] Kajikiya, R., Necessary and sufficient condition for existence and uniqueness of nodal solutions to sublinear elliptic equations, Adv. Differential Equations, 6, 1317-1346 (2001) · Zbl 1010.35040
[11] Kaper, H.; Knaap, G.; Kwong, M. K., Existence theorems for second order boundary value problems, Differential Integral Equations, 4, 543-554 (1991) · Zbl 0732.34019
[12] Kitano, M.; Kusano, T., On a class of second order quasilinear ordinary differential equations, Hiroshima Math. J., 25, 321-355 (1995) · Zbl 0835.34034
[13] Kusano, T.; Naito, M., Sturm-Liouville eigenvalue problems from half-linear ordinary differential equations, Rocky Mountain J. Math., 31, 1039-1054 (2001) · Zbl 1006.34025
[14] Lee, Y.-H.; Sim, I., Global bifurcation phenomena for singular one-dimensional \(p\)-Laplacian, J. Differential Equations, 229, 229-256 (2006) · Zbl 1113.34010
[15] Y.-H. Lee, I. Sim, Existence results of sign-changing solutions for singular one-dimensional \(p\); Y.-H. Lee, I. Sim, Existence results of sign-changing solutions for singular one-dimensional \(p\) · Zbl 1138.34010
[16] Li, H. J.; Yeh, C. C., Sturmian comparison theorem for half-linear second-order differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 125, 1193-1204 (1995) · Zbl 0873.34020
[17] Ma, R.; Thompson, B., Nodal solutions for nonlinear eigenvalue problems, Nonlinear Anal., 59, 707-718 (2004) · Zbl 1059.34013
[18] Ma, R.; Thompson, B., Multiplicity results for second-order two-point boundary value problems with superlinear or sublinear nonlinearities, J. Math. Anal. Appl., 303, 726-735 (2005) · Zbl 1075.34017
[19] Ma, R.; Thompson, B., A note on bifurcation from an interval, Nonlinear Anal., 62, 743-749 (2005) · Zbl 1074.34029
[20] Manasevich, R.; Njoku, F. I.; Zanolin, F., Positive solutions for the one-dimensional \(p\)-Laplacian, Differential Integral Equations, 8, 213-222 (1995) · Zbl 0815.34015
[21] Manasevich, R.; Zanolin, F., Time-mappings and multiplicity of solutions for the one-dimensional \(p\)-Laplacian, Nonlinear Anal., 21, 269-291 (1993) · Zbl 0792.34021
[22] Naito, Y.; Tanaka, S., On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations, Nonlinear Anal., 56, 919-935 (2004) · Zbl 1046.34038
[23] Reichel, W.; Walter, W., Radial solutions of equations and inequalities involving the \(p\)-Laplacian, J. Inequal. Appl., 1, 47-71 (1997) · Zbl 0883.34024
[24] Wang, J., The existence of positive solutions for the one-dimensional \(p\)-Laplacian, Proc. Amer. Math. Soc., 125, 2275-2283 (1997) · Zbl 0884.34032
[25] Wang, Z.; Zhang, J., Positive solutions for one-dimensional \(p\)-Laplacian boundary value problems with dependence on the first order derivative, J. Math. Anal. Appl., 314, 618-630 (2006) · Zbl 1094.34016
[26] Zhang, M., Nonuniform nonresonance of semilinear differential equations, J. Differential Equations, 166, 33-50 (2000) · Zbl 0962.34062
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.