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Global Preservation of Nodal Structure in Coupled Systems of Nonlinear Sturm-Liouville Boundary Value Problems

Robert Stephen Cantrell
Proceedings of the American Mathematical Society
Vol. 107, No. 3 (Nov., 1989), pp. 633-644
DOI: 10.2307/2048159
Stable URL: http://www.jstor.org/stable/2048159
Page Count: 12
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Global Preservation of Nodal Structure in Coupled Systems of Nonlinear Sturm-Liouville Boundary Value Problems
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Abstract

In this paper, we examine the solution set to the coupled system \begin{equation*}\tag{*} \begin{cases} -(p_1(x)u'(x))' + q_1(x)u(x) = \lambda u(x) + u(x) \cdot f(u(x), v(x)) \\ -(p_2(x)v'(x))' + q_2(x)v(x) = \mu v(x) + v(x) \cdot g(u(x), v(x)),\end{cases} \end{equation*} where λ, μ ∈ R, x ∈ [ a, b ], and the system (*) is subject to zero Dirichlet boundary data on u and v. 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), v 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 (λ, μ, u, 0) with u having n - 1 simple zeros in (a, b) to solutions of (*) of the form (λ, μ, 0, v) with v having m - 1 simple zeros in (a, b).

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