On the number of positive solutions of nonlinear systems. (English) Zbl 1036.34032

This paper is devoted to the existence, multiplicity and nonexistence of positive solutions to boundary value problems on \([0,1]\) for a class of second-order differential systems where the main common operator is the one-dimensional \(p\)-Laplacian, \(p> 1\). The author uses a fixed-point theorem in a cone due to M. A. Krasnoselskij [Positive solutions of operator equations. Groningen: The Netherlands: P. Noordhoff Ltd. (1964; Zbl 0121.10604)].


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations


Zbl 0121.10604
Full Text: DOI


[1] Agarwal, R.; Lu, H.; O’Regan, D., Eigenvalues and the one-dimensional \(p\)-Laplacian, J. Math. Anal. Appl., 266, 383-400 (2002) · Zbl 1002.34019
[2] Bandle, C.; Coffman, C. V.; Marcus, M., Nonlinear elliptic problems in annular domains, J. Differential Equations, 69, 322-345 (1987) · Zbl 0618.35043
[3] Ben-Naoum, A.; De Coster, C., On the existence and multiplicity of positive solutions of the \(p\)-Laplacian separated boundary value problem, Differential Integral Equations, 10, 1093-1112 (1997) · Zbl 0940.35086
[4] Deimling, K., Nonlinear Functional Analysis (1985), Springer: Springer Berlin · Zbl 0559.47040
[5] 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
[6] Dunninger, D.; Wang, H., Multiplicity of positive radial solutions for an elliptic system on an annulus, Nonlinear Anal., 42, 803-811 (2000) · Zbl 0959.35051
[7] Dunninger, D.; Wang, H., Existence and multiplicity of positive radial solutions for elliptic systems, Nonlinear Anal., 29, 1051-1060 (1997) · Zbl 0885.35028
[8] Erbe, L.; Hu, S.; Wang, H., Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl., 184, 640-648 (1994) · Zbl 0805.34021
[9] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press Orlando, FL · Zbl 0661.47045
[10] Krasnoselskii, M., Positive Solutions of Operator Equations (1964), Noordhoff: Noordhoff Groningen
[11] Lin, S. S., On the existence of positive radial solutions for semilinear elliptic equations in annular domains, J. Differential Equations, 81, 221-233 (1989) · Zbl 0691.35036
[12] Manasevich, R.; Mawhin, J., The spectrum of \(p\)-Laplacian systems with various boundary conditions and applications, Adv. Differential Equations, 5, 1289-1318 (2000) · Zbl 0992.34063
[13] Wang, H., On the existence of positive solutions for semilinear elliptic equations in the annulus, J. Differential Equations, 109, 1-7 (1994) · Zbl 0798.34030
[16] Wang, J., The existence of positive solutions for the one-dimensional \(p\)-Laplacian, Proc. Amer. Math. Soc., 125, 2275-2283 (1997) · Zbl 0884.34032
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