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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Global preservation of nodal structure in coupled systems of nonlinear Sturm-Liouville boundary value problems


Author: Robert Stephen Cantrell
Journal: Proc. Amer. Math. Soc. 107 (1989), 633-644
MSC: Primary 34B25; Secondary 58F19, 92A15
MathSciNet review: 975633
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Abstract: In this paper, we examine the solution set to the coupled system ($ *$)

$\displaystyle \left\{ {_{ - ({p_2}(x)\upsilon '(x))' + {q_2}(x)\upsilon (x) = \... ...(x))' + {q_1}(x)u(x) = \lambda u(x) + u(x) \cdot f(u(x),\upsilon (x))}} \right.$

where $ \lambda ,\mu \in R,x \in [a,b]$, and the system ($ *$) is subject to zero Dirichlet boundary data on $ u$ and $ \upsilon $. We determine conditions on $ f$ and $ g$ which permit us to assert the existence of continua of solutions to ($ *$) characterized by $ u$ having $ n - 1$ simple zeros in $ (a,b),\upsilon $ having $ m - 1$ simple zeros in $ (a,b)$, where $ n$ and $ m$ are positive but not necessarily equal integers. Moreover, we also determine conditions under which these continua link solutions to ($ *$) of the form $ (\lambda ,\mu ,u,0)$ with $ u$ having $ n - 1$ simple zeros in $ (a,b)$ to solutions of ($ *$) of the form $ (\lambda ,\mu ,0,\upsilon )$ with $ \upsilon $ having $ m - 1$ simple zeros in $ (a,b)$.

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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0975633-X
PII: S 0002-9939(1989)0975633-X
Article copyright: © Copyright 1989 American Mathematical Society