Disconjugacy conditions and spectrum structure of clamped beam equations with two parameters

Yanqiong Lu; Ruyun Ma

doi:10.3934/cpaa.2020145

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2020, Volume 19, Issue 6: 3283-3302. Doi: 10.3934/cpaa.2020145

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Disconjugacy conditions and spectrum structure of clamped beam equations with two parameters

Yanqiong Lu^, and
Ruyun Ma

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

^* Corresponding author
^* Corresponding author

Received: July 31, 2019

Revised: December 31, 2019

Published: March 2020

The first author is supported by National Natural Science Foundation of China (No.11901464, No.11671322, No.11801453), Gansu provincial National Science Foundation of China (No.1606RJYA232) and NWNU-LKQN-15-16

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Abstract

In this work, we apply the 'disconjugacy theory' and Elias's spectrum theory to study the disconjugacy $ u^{(4)} + \beta u''-\alpha u = 0 $ with two parameters $ \alpha,\beta\in\mathbb{R} $ and the spectrum structure of the linear operator $ u^{(4)} + \beta u''-\alpha u $ coupled with the clamped beam conditions $ u(0) = u'(0) = u(1) = u'(1) = 0 $. As the application of our results, we obtain the global structure of nodal solutions of the corresponding nonlinear analogue based on the bifurcation theory.

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Mathematics Subject Classification: Primary: 34B15; Secondary: 34B18.

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