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

Korman, Philip

doi:10.1090/proc/12905

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

Proc. Amer. Math. Soc. 144 (2016), 2539-2546

DOI: https://doi.org/10.1090/proc/12905

Published electronically: October 22, 2015

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Abstract:

We prove that positive solutions of the two-point boundary value problem \[ u''(x)+\lambda f(u(x))=0,\;\; \mbox {for $-1<x<1$},\;\; u(-1)=u(1)=0,\] 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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