Nontrivial solutions of singular two-point boundary value problems with sign-changing nonlinear terms. (English) Zbl 1111.34019

Summary: The singular two-point boundary value problem \[ -u''(t)=h(t)f(u(t),\;t\in(0,1);\quad u(0)=u(1)=0, \] is considered under some conditions concerning the first eigenvalue corresponding to the relevant linear problem, where \(h\) is allowed to be singular at both \(t=0\) and \(t=1\). Moreover, \(f:(-\infty,+\infty) \to(-\infty,+\infty)\) is a sign-changing function and not necessarily bounded from below. By computing the topological degree of an completely continuous field, existence results for nontrivial solutions are established.


34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] 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
[2] Cheng, J., Positive solutions of second order boundary value problems, Acta Math. Sinica, 44, 429-436 (2001), (in Chinese) · Zbl 1018.34020
[3] Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0559.47040
[4] Erbe, L. H.; Mathsen, R. M., Positive solutions for singular nonlinear boundary value problems, Nonlinear Anal., 46, 979-986 (2001) · Zbl 1007.34020
[5] Guo, Dajun, Nonlinear Functional Analysis (1985), Shandong Sci. Tech.: Shandong Sci. Tech. Ji’nan, (in Chinese) · Zbl 0636.45012
[6] Guo, Dajun; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press · Zbl 0661.47045
[7] Guo, Dajun; Sun, Jingxian, Nonlinear Integral Equations (1987), Shandong Sci. Tech.: Shandong Sci. Tech. Ji’nan, (in Chinese)
[8] 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
[9] Li, Fuyi; Han, Guodong, Existence of non-zero solutions to nonlinear Hammerstein integral equation, J. Shanxi Univ. Nat. Sci. Ed., 26, 283-286 (2003)
[10] Liu, Jiaquan, Positive solutions for singular boundary problem of second order, J. Qufu Norm. Univ. Nat. Sci. Ed., 28, 1-7 (2002), (in Chinese) · Zbl 1036.34031
[11] Ma, R., Positive solutions of singular second order boundary value problem, Acta Math. Sinica, 41, 1225-1230 (1998), (in Chinese) · Zbl 1027.34025
[12] O’Regan, D., Theory of Singular Boundary Value Problems (1994), World Scientific: World Scientific Singapore · Zbl 0808.34022
[13] O’Regan, D., Singular Dirichlet boundary value problems-I, superlinear and nonresonant case, Nonlinear Anal., 29, 221-245 (1997) · Zbl 0884.34028
[14] Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1967), Prentice Hall: Prentice Hall New York · Zbl 0153.13602
[15] Sun, Jingxian, Non-zero solutions to superlinear Hammerstein integral equations and applications, Chinese Ann. Math. Ser. A, 7, 528-535 (1986), (in Chinese) · Zbl 0633.45006
[16] Sun, Jingxian; Zhang, Guowei, Nontrivial solutions of singular superlinear Sturm-Liouville problems, J. Math. Anal. Appl., 313, 518-536 (2006) · Zbl 1100.34019
[17] Wang, J., On positive solutions of singular nonlinear two-point boundary value problems, J. Differential Equations, 107, 163-174 (1994) · Zbl 0792.34023
[18] 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
[19] Zhang, Y., Positive solutions of singular sublinear Emden-Fowler boundary value problems, J. Math. Anal. Appl., 185, 215-222 (1994) · Zbl 0823.34030
[20] 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
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