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

Han, Guodong; Wu, Ying

J. Math. Anal. Appl. 325, No. 2, 1327-1338 (2007).

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.

MSC:

34B16	Singular nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

singular boundary value problems; nontrivial solution; cone; Leray-Schauder degree

References:

[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