The existence of countably many positive solutions for one-dimensional \(p\)-Laplacian with infinitely many singularities on the half-line. (English) Zbl 1157.34016

The authors study the existence of countable many positive solutions of nonlinear \(m\)-point boundary value problem with a \(p\)-Laplacian operator on a half-line of the form
\[ (\phi_p(u'))'+a(t) f(u(t))=0, \quad t\in (0,\infty), \]
\[ u(0)=\sum^{m-2}_{i=1}\alpha_i u(\xi_i), \quad u'(\infty)=0, \]
where \(\phi_p(s)=| s| ^{p-2}s\), \(p>1\), \(\xi_i\in (0,\infty)\) with \(0<\xi_1<\xi_2<\cdots<\xi_{m-2}<+\infty\), \(\alpha_i\in (0,\infty)\) with \(0<\sum^{m-2}_{i=1} \alpha_i<1\), \(f\in C([0,\infty), [0,\infty))\), \(a\in C([0,\infty), [0,\infty))\)
may be singular at countable many points in \([1,\infty)\). The main tool they used is the fixed point index theory in cones.
Reviewer: Ruyun Ma (Lanzhou)


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
Full Text: DOI


[1] Agarwal, R.P.; Regan, D.O., Infinite interval problems for differential difference and integral equations, (2001), Kluwer Academic Dordrecht
[2] Liu, B.F.; Zhang, J.H., The existence of positive solutions for some nonlinear boundary value problems with linear mixed boundary conditions, J. math. anal. appl., 309, 505-516, (2005) · Zbl 1086.34022
[3] Aronson, D.; Crandall, M.G.; Peletier, L.A., Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear anal., 6, 1001-1022, (1982) · Zbl 0518.35050
[4] Baxley, J.V., Existence and uniqueness of nonlinear boundary value problems on infinite intervals, J. math. anal. appl., 147, 1133-1274, (1990)
[5] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag New York · Zbl 0559.47040
[6] Kaufmann, E.R.; Kosmatov, N., A multiplicity result for a boundary value problem with infinitely many singularities, J. math. anal. appl., 269, 444-453, (2002) · Zbl 1011.34012
[7] Liu, B., Positive solutions three-points boundary value problems for one-dimensional p-Laplacian with infinitely many singularities, Appl. math. lett., 17, 655-661, (2004) · Zbl 1060.34006
[8] Su, H.; Wei, Z.L.; Xu, F.Y., The existence of countably many positive solutions for a system of nonlinear singular boundary value problems with the p-Laplacian operator, J. math. anal. appl., 325, 319-332, (2007) · Zbl 1108.34015
[9] Wang, Y.Y.; Hou, C., Existence of multiple positive solutions for one-dimensional p-Laplacian, J. math. anal. appl., 315, 144-153, (2006) · Zbl 1098.34017
[10] Ma, R.Y.; Zhang, J.H.; Fu, S.M., The method of lower and upper solutions for fourth-order two-point boundary value problems, J. math. anal. appl., 215, 415-422, (1997) · Zbl 0892.34009
[11] Iffland, G., Positive solutions of a problem Emden cfowler type with a type free boundary, SIAM J. math. anal., 18, 283-292, (1987) · Zbl 0637.34013
[12] Bai, C.; Fang, J., Existence of multiple positive solution for nonlinear m-point boundary value problems, Appl. math. comput., 140, 297-305, (2003) · Zbl 1033.34019
[13] Kawano, N.; Yanagida, E.; Yotsutani, S., Structure theorems for positive radial solutions to \(\mathit{Lu} + K(| x |) u^p = 0\) in rn, Funkcial. ekvac., 36, 557-579, (1993) · Zbl 0793.34024
[14] Wang, J.Y., The existence of positive solutions for the one-dimensional p-Laplacian, Proc. am. math. soc., 125, 2275-2283, (1997) · Zbl 0884.34032
[15] Yan, B.Q., Multiple unbounded solutions of boundary value problems for second-order differential equations on the half-line, Nonlinear anal., 51, 1031-1044, (2002) · Zbl 1021.34021
[16] Zima, M., On positive solution of boundary value problems on the half-line, J. math. anal. appl., 259, 127-136, (2001) · Zbl 1003.34024
[17] Lian, H., Triple positive solutions for boundary value problems on infinite intervals, Nonlinear anal., 67, 2199-2207, (2007) · Zbl 1128.34011
[18] Liu, Y.S., Existence and unboundedness of positive solutions for singular boundary value problems on half-line, Appl. math. comput., 1404, 543-556, (2003) · Zbl 1036.34027
[19] Ren, J.L.; Ge, W.G.; Ren, B.X., Existence of positive solutions for quasi-linear boundary value problems, Acta math. appl. sinica, 21, 3, 353-358, (2005), (in Chinese) · Zbl 1113.34016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.