Sturm type problems for singular \(p\)-Laplacian boundary value problems. (English) Zbl 1035.34013
The author considers the following singular boundary value problem for the \(p\)-Laplacian operator: \(L_{p}u+\lambda c(t)u^{ (p-1)}=0\), \(0<t<1\), \(u(0)=u(1)=0\).
Here, \(L_{p}u(t)=(a(t)u^{(p-1)})'\). The functions \(a^{-1}\) and \(c\) may be singular at t=0 and/or t=1. It is shown that there exists an increasing sequence of eigenvalues with a description of the corresponding eigenfunctions. As an application the author proves the existence of at least one solution for the nonresonance problem: \(L_{p}u(t)+f(t,u)=0\), \(u(0)=u(1)=0\).
Here, \(L_{p}u(t)=(a(t)u^{(p-1)})'\). The functions \(a^{-1}\) and \(c\) may be singular at t=0 and/or t=1. It is shown that there exists an increasing sequence of eigenvalues with a description of the corresponding eigenfunctions. As an application the author proves the existence of at least one solution for the nonresonance problem: \(L_{p}u(t)+f(t,u)=0\), \(u(0)=u(1)=0\).
Reviewer: Abdelkader Boucherif (Dhahran)
MSC:
| 34B24 | Sturm-Liouville theory |
| 34B16 | Singular nonlinear boundary value problems for ordinary differential equations |
References:
| [1] | Mirzov, J. D., On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems, J. Math. Anal. Appl., 53, 418-425 (1976) · Zbl 0327.34027 |
| [2] | Del Pino, M.; Elgueta, M.; Manasevich, R., A homotopic deformation along \(p\) of a Leray-Schauder degree result and existence for \((|u^′|^{p−2}u^′)^′\)+f(t,u)=0,u(0)=u(T)=0,p>1\), J. Differential Equations, 80, 1-13 (1989) · Zbl 0708.34019 |
| [3] | Walter, W., Stwrm-Liouville theory for the radial \(Δ_p\)-operator, Math. Z., 227, 175-185 (1998) · Zbl 0915.34022 |
| [4] | Reichel, W.; Walter, W., Sturm-Liouville type problems for the \(p\)-Laplacian under asymptotic non-resonance conditions, J. Differential Equations, 156, 50-70 (1999) · Zbl 0931.34059 |
| [5] | Liu, X., A note on the Sturmian theorem for singular boundary value problems, J. Math. Anal. Appl., 237, 393-403 (1999) · Zbl 0936.34021 |
| [6] | Asakawa, H., Nonresonant singular two-point boundary value problems, Nonlinear Anal., 44, 791-809 (2001) · Zbl 0992.34011 |
| [7] | Reid, W. T., Sturmian Theory for Ordinary Differential Equations (1980), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0459.34001 |
| [8] | Boccardo, L.; Drabek, P.; Giachetti, D.; Kucera, M., Generalization of Fredholm alternative for nonlinear differential operators, Nonlinear Anal., 10, 1083-1103 (1986) · Zbl 0623.34031 |
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