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Non-singular solutions of two-point problems, with multiple changes of sign in the nonlinearity


Author: Philip Korman
Journal: Proc. Amer. Math. Soc. 144 (2016), 2539-2546
MSC (2010): Primary 34B15
DOI: https://doi.org/10.1090/proc/12905
Published electronically: October 22, 2015
MathSciNet review: 3477070
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Abstract: We prove that positive solutions of the two-point boundary value problem

$\displaystyle u''(x)+\lambda f(u(x))=0,\;\;$$\displaystyle \mbox {for $-1<x<1$},\;\; u(-1)=u(1)=0,$

satisfying $ \max u=u(0)>\gamma $, are non-singular, provided that $ f(u)$ is predominantly negative for $ u \in (0,\gamma ]$, and superlinear for $ u>\gamma $. This result adds a solution curve without turns to whatever is known about the solution set for $ u(0) \in (0,\gamma )$. In particular, we combine it with the well-known cases of parabola-like, or $ S$-shaped solution curves.

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Additional Information

Philip Korman
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025
Email: kormanp@ucmail.uc.edu

DOI: https://doi.org/10.1090/proc/12905
Keywords: Global solution curves, non-singular positive solutions
Received by editor(s): October 27, 2014
Received by editor(s) in revised form: July 14, 2015
Published electronically: October 22, 2015
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society