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1997 Existence and multiplicity of solutions of some conservative pendulum-type equations with homogeneous Dirichlet conditions
A. Cañada, F. Roca
Differential Integral Equations 10(6): 1113-1122 (1997).

Abstract

Let us consider the resonant boundary value problem $$ \begin{align} - &u''(x) - u(x) + g(u(x)) = h(x), \quad x \in [0,\pi], \\ &u(0) = u(\pi) = 0, \end{align} $$ where $ g: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous and $ T$-periodic function with zero mean value, not identically zero, and $ h \in C[0,\pi].$ If each $ h \in C[0,\pi] $ is written as $ h(x) = a\sin x + \tilde{h}(x), $ where $ a \in \mathbb{R} $ and $\int_{0}^{\pi} \tilde{h}(x) \sin x \ dx = 0,$ then, it is shown that for each $ \tilde{h},$ there are real numbers $ a_{1}(\tilde{h}) < 0 < a_{2}(\tilde{h})$ (which depend continuously on $ \tilde{h}$), such that there is solution if and only if $ a\in [a_{1}(\tilde{h}),a_{2}(\tilde{h})].$ In relation to the multiplicity, it is proved that the number of solutions increases to infinity as $ a $ goes to zero. The proof combines different tools such as Liapunov-Schmidt reduction and upper-lower solutions notions, together with a careful analysis of the connected subsets of the solution set of the auxiliary equation in the alternative method, as well as a detailed study of the oscillatory behavior of some integrals associated to the bifurcation equation of the previous problem.

Citation

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A. Cañada. F. Roca. "Existence and multiplicity of solutions of some conservative pendulum-type equations with homogeneous Dirichlet conditions." Differential Integral Equations 10 (6) 1113 - 1122, 1997.

Information

Published: 1997
First available in Project Euclid: 1 May 2013

zbMATH: 0938.34013
MathSciNet: MR1608041

Subjects:
Primary: 34B15
Secondary: 34C25, 47H15, 47N20

Rights: Copyright © 1997 Khayyam Publishing, Inc.

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Vol.10 • No. 6 • 1997
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