Twin positive solutions for the one-dimensional \(p\)-Laplacian boundary value problems. (English) Zbl 1061.34013

The authors deal with the one-dimensional \(p\)-Laplacian \[ (g(u'))'+ e(t) f(u)= 0 \] subject to one of the following three pairs of nonlinear boundary conditions:
a) \(u(0)= B_0(u'(0))= 0\), \(u(1)+ B_1(u'(1))= 0\),
b) \(u(0)= B_0(u'(0))= 0\), \(u'(1)= 0\),
c) \(u'(0)= 0\), \(u(1)+ B_1(u'(1))= 0\), where \(g(v):= |v|^{p-2} v\), \(p> 1\).
The authors show the existence of at least two positive solutions. To this end, they use a fixed-point theorem in cones.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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[1] Avery, R.I.; Chyan, C.J.; Henderson, J., Twin solutions of boundary value problems for ordinary differential equations and finite difference equations, Comput. math. appl., 42, 695-704, (2001) · Zbl 1006.34022
[2] Avery, R.I.; Henderson, J., Two positive fixed points of nonlinear operators on ordered Banach spaces, Commun. appl. nonlinear anal., 8, 27-36, (2001) · Zbl 1014.47025
[3] Erbe, L.H.; Hu, S.C.; Wang, H.Y., Multiple positive solutions of some boundary value problems, J. math. anal. appl., 184, 640-648, (1994) · Zbl 0805.34021
[4] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press Boston · Zbl 0661.47045
[5] Henderson, J.; Thompson, H.B., Multiple symmetric positive solutions for second order boundary value problem, Proc. amer. math. soc., 128, 2373-2379, (2000) · Zbl 0949.34016
[6] Kong, L.; Wang, J., Multiple positive solutions for the one-dimensional p-Laplacian, Nonlinear anal., 42, 1327-1333, (2000) · Zbl 0961.34012
[7] Lian, W.C.; Wong, F.H.; Yech, C.C., On the existence of positive solutions of nonlinear second order differential equations, Proc. amer. math. soc., 124, 1117-1126, (1996) · Zbl 0857.34036
[8] 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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