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Nodal solutions of second order nonlinear boundary value problems

Published online by Cambridge University Press:  01 May 2009

LINGJU KONG  and
QINGKAI KONG
LINGJU KONG
Affiliation:
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, U.S.A. e-mail: lingju-kong@utc.edu
QINGKAI KONG
Affiliation:
Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, U.S.A. e-mail: kong@math.niu.edu
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Abstract

We study the nonlinear boundary value problem (BVP) consisting of the equation −(p(t)y′)′ + q(t)y = w(t)f(y) on [a, b] and a general separated boundary condition (BC). By comparing it with a linear Sturm--Liouville problem (SLP) we obtain conditions for the existence and nonexistence of nodal solutions of this problem. More specifically, let λn, n = 0, 1, 2, . . ., be the nth eigenvalue of the corresponding linear SLP. Then the BVP has a pair of solutions with exactly n zeros in (a, b) if λn is in the interior of the range of f(y)/y; and does not have any solution with exactly n zeros in (a, b) if λn is outside this range. These conditions become necessary and sufficient when f(y)/y is monotone on (−∞, 0) and on (0, ∞). We also discuss the changes of the number of different types of nodal solutions as the equation or the BC changes. Our results are obtained without assuming the global existence and uniqueness of solutions of the corresponding initial value problems.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 3 , May 2009 , pp. 747 - 763
Copyright
Copyright © Cambridge Philosophical Society 2008

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