zbMATH — the first resource for mathematics

Exact number of positive solutions for a class of semipositone problems. (English) Zbl 1028.34027
The author considers the nonlinear eigenvalue problem \[ y''(t)= \lambda(y^p(t)- y^q(t)),\quad 0< t< 1,\qquad y(-1)= y(1)= 0, \] where \(p> q> -1\), and \(\lambda> 0\) is a positive parameter. He gives a complete study to the exact number of positive solutions to the above problem. The proofs are based upon the time-maps estimate.
Reviewer: Ruyun Ma (Lanzhou)

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47J15 Abstract bifurcation theory involving nonlinear operators
Full Text: DOI
[1] Castro, A.; Gadam, S.; Shivaji, R., Evolution of positive solution curves in semipositone problems with concave nonlinearities, J. math. anal. appl., 245, 282-293, (2000) · Zbl 0985.34014
[2] Davidson, F.A.; Rynne, B.P., Asymptotic oscillations of continua of positive solutions of a semilinear sturm – liouville problem, J. math. anal. appl., 252, 617-630, (2000) · Zbl 0974.34023
[3] Fu, C.; Lin, S., Uniqueness of positive solutions for semilinear elliptic equations on annular domains, Nonlinear anal., 44, 749-758, (2001) · Zbl 0996.34018
[4] Liu, Z., Exact number of solutions of a class of two-point boundary value problems involving concave and convex nonlinearities, Nonlinear anal., 46, 181-197, (2001) · Zbl 1003.34020
[5] Liu, Z.; Zhang, X., A class of two-point boundary value problems, J. math. anal. appl., 254, 599-617, (2001) · Zbl 0983.34010
[6] Ouyang, T.; Shi, J., Exact multiplicity of positive solutions for a class of semilinear problem II, J. differential equations, 158, 94-151, (1999) · Zbl 0947.35067
[7] Wang, S.; Long, D., An exact multiplicity theorem involving concave – convex nonlinearities and its application to stationary solutions of a singular diffusion problem, Nonlinear anal., 44, 469-486, (2001) · Zbl 0992.34014
[8] Yadava, S.L., Uniqueness of positive radial solutions of the Dirichlet problems −δu=up±uq in an annulus, J. differential equations, 139, 194-217, (1997) · Zbl 0884.34026
[9] Zhao, P.; Zhong, C., On the infinitely many positive solutions of a supercritical elliptic problem, Nonlinear anal., 44, 123-139, (2001) · Zbl 0970.35040
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.