×

Nodal solutions for nonlinear eigenvalue problems. (English) Zbl 1059.34013

Summary: We are concerned with determining values of \(\lambda\), for which there exist nodal solutions of the boundary value problem \[ u''+ ra(t) f(u)= 0,\quad 0< t< 1,\quad u(0)= u(1)= 0. \] The proof of our main result is based upon bifurcation techniques.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ambrosetti, A.; Hess, P., Positive solutions of asymptotically linear elliptic eigenvalue problems, J. math. anal. appl. (, 73), 2, 411-422, (1980) · Zbl 0433.35026
[2] Elias, U., Eigenvalue problems for the equation \(L_y + \lambda_p(x) y = 0\), J. diff. equations, 29, 28-57, (1978)
[3] Erbe, L.H.; Haiyan Wang, On the existence of positive solutions of ordinary differential equations, Proc. amer. math. soc., 120, 3, 743-748, (1994) · Zbl 0802.34018
[4] Henderson, J.; HaiyanWang, Positive solutions for nonlinear eigenvalue problems, J. math. anal. appl., 208, 252-259, (1997) · Zbl 0876.34023
[5] Naito, Y.; Tanaka, S., On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations, Nonlinear anal. TMA, 56, 4, 919-935, (2004) · Zbl 1046.34038
[6] Rabinowitz, P.H., Nonlinear Sturm-Liouville problems for second order ordinary differential equations, Commun. pure appl. math., 23, 939-961, (1970) · Zbl 0202.08902
[7] Rabinowitz, P.H., Some global results for nonlinear eigenvalue problems, J. funct. anal., 7, 487-513, (1971) · Zbl 0212.16504
[8] Ruf, B.; Srikanth, P.N., Multiplicity results for ODEs with nonlinearities crossing all but a finite number of eigenvalues, Nonlinear anal. TMA, 10, 2, 157-163, (1986) · Zbl 0586.34017
[9] Rynne, B.P., Infinitely many solutions of superlinear fourth order boundary value problems, Topol. methods nonlinear anal., 19, 2, 303-312, (2002) · Zbl 1017.34015
[10] Walter, W., Ordinary differential equations, (1998), Springer New York
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.