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Nontrivial solutions of singular superlinear Sturm–Liouville problems. (English) Zbl 1100.34019

The authors study the singular superlinear problem
\[ -(p(x)y')'-q(x)y=h(x)f(y),\quad 0<x<1, \]
\[ \alpha_1 y(0)+\beta_1 y'(0)=0,\quad \alpha_2 y(1)+\beta_2 y'(1)=0. \]
The function \(h\) is allowed to be singular at both \(x=0\) and \(x=1\). In addition, \(f\) is not assumed to be nonnegative. The assumption of nonnegativity of \(f\) has been very often required in the existing literature. Omitting this condition requires a different approach. Using topological degree theory, the authors establish conditions guaranteeing the existence of nontrivial solutions and positive solutions to the above boundary value problem. A nonsingular case is discussed as well.
Reviewer: Pavel Rehak (Brno)

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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[1] O’Regan, D., Theory of singular boundary value problems, (1994), World Scientific Singapore · Zbl 0808.34022
[2] Habets, P.; Zanolin, F., Upper and lower solutions for a generalized emden – fowler equation, J. math. anal. appl., 181, 684-700, (1994) · Zbl 0801.34029
[3] Erbe, L.H.; Mathsen, R.M., Positive solutions for singular nonlinear boundary value problems, Nonlinear anal., 46, 979-986, (2001) · Zbl 1007.34020
[4] Wang, J., On positive solutions of singular nonlinear two-point boundary value problems, J. differential equations, 107, 163-174, (1994) · Zbl 0792.34023
[5] Zhang, Y., Positive solutions of singular sublinear emden – fowler boundary value problems, J. math. anal. appl., 185, 215-222, (1994) · Zbl 0823.34030
[6] Ma, R., Positive solutions of singular second order boundary value problem, Acta math. sinica, 41, 1225-1230, (1998), (in Chinese) · Zbl 1027.34025
[7] Wei, Z., Positive solutions of singular boundary value problems of negative exponent emden – fowler equations, Acta math. sinica, 41, 655-662, (1998), (in Chinese) · Zbl 1027.34024
[8] Zhao, Z., Positive solutions of boundary value problems for nonlinear singular differential equations, Acta math. sinica, 43, 179-188, (2000), (in Chinese) · Zbl 1018.34018
[9] Cheng, J., Positive solutions of second order boundary value problems, Acta math. sinica, 44, 429-436, (2001), (in Chinese) · Zbl 1018.34020
[10] Agarwal, R.P.; O’Regan, D., A note on existence of nonnegative solutions to singular semi-positone problems, Nonlinear anal., 36, 615-622, (1999) · Zbl 0921.34027
[11] Cheng, J., Positive solutions of singular semi-positone problems, Acta math. sinica, 44, 673-678, (2001), (in Chinese) · Zbl 1024.34017
[12] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press San Diego, CA · Zbl 0661.47045
[13] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin · Zbl 0559.47040
[14] Guo, D.; Sun, J.; Liu, Z., Functional methods in nonlinear ordinary differential equations, (1995), Shandong Sci. Tech. Jinan, (in Chinese)
[15] Krasnoselskii, M.A.; Zabreiko, P.P., Geometrical methods of nonlinear analysis, (1984), Springer-Verlag New York
[16] Rubinstein, Z., A course in ordinary and partial differential equations, (1969), Academic Press New York · Zbl 0175.37801
[17] Guo, D.; Sun, J., Nonlinear integral equations, (1987), Shandong Sci. Tech. Jinan, (in Chinese)
[18] Krasnoselskii, M.A., Topological methods in the theory of nonlinear integral equations, (1964), Pergamon Press Oxford
[19] Protter, M.H.; Weinberger, H.F., Maximum principles in differential equations, (1967), Prentice Hall New York · Zbl 0153.13602
[20] Li, Y., Positive solutions of second-order boundary value problems with sign-changing nonlinear terms, J. math. anal. appl., 282, 232-240, (2003) · Zbl 1030.34023
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