Existence of nodal solutions of a nonlinear eigenvalue problem with indefinite weight function. (English) Zbl 1173.34310

Summary: We investigate the existence of nodal solutions of the indefinite weight boundary value problem \[ u^{\prime \prime }+rh(t)f(u)= 0 , \quad 0 < t < 1,\qquad u(0)=u(1)=0, \] where \(h\in C\)[0,1] changes sign. The proof of our main result is based upon bifurcation techniques.


34B15 Nonlinear boundary value problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
Full Text: DOI


[1] Henderson, J.; Wang, H. Y., Positive solutions for nonlinear eigenvalue problems, J. Math. Anal. Appl., 208, 252-259 (1997) · Zbl 0876.34023
[2] Ma, R. Y.; Thompson, B., Nodal solutions for nonlinear eigenvalue problems, Nonlinear Anal., 59, 707-718 (2004) · Zbl 1059.34013
[3] Rabinowitz, P. H., Nonlinear Sturm-Liouville problems for second order ordinary differential equations, Commun. Pure Appl. Math., 23, 939-961 (1970) · Zbl 0202.08902
[4] Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7, 487-513 (1971) · Zbl 0212.16504
[5] Binding, P.; Ye, Q., Variational principles for indefinite eigenvalue problems, Linear Algebra Appl., 218, 251-262 (1995) · Zbl 0821.15005
[6] Hai, D. D., Positive solutions to a class of elliptic boundary value problems, J. Math. Anal. Appl., 227, 195-199 (1998) · Zbl 0915.35043
[7] Brown, K. J.; Lin, S. S., On the existence of positive eigenfunction for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75, 112-120 (1980) · Zbl 0437.35058
[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] Ince, E. L., Ordinary Differential Equations (1926), Dover: Dover New York · Zbl 0063.02971
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.