Classification of bifurcation curves of positive solutions for a nonpositone problem with a quartic polynomial

Kuan-Ju  Huang; Yi-Jung  Lee; Tzung-Shin Yeh

doi:10.3934/cpaa.2016.15.1497

Article Contents

2016, Volume 15, Issue 4: 1497-1514. Doi: 10.3934/cpaa.2016.15.1497

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Classification of bifurcation curves of positive solutions for a nonpositone problem with a quartic polynomial

1.
Department of Applied Mathematics, National University of Tainan, Tainan 700

Received: August 31, 2015

Revised: January 31, 2016

Published: June 2016

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Abstract

We study exact multiplicity and bifurcation curves of positive solutions of the boundary value problem \begin{eqnarray} &u"(x)+\lambda (-u^4+\sigma u^3-\tau u^2+\rho u)=0, -1 < x < 1, \\ &u(-1)=u(1)=0, \end{eqnarray} where $\sigma, \tau \in \mathbb{R}$, $\rho \geq 0,$ and $\lambda >0$ is a bifurcation parameter. Then on the $(\lambda, \|u\|_\infty)$-plane, we give a classification of four qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, a $\subset$-shaped curve and a monotone increasing curve.

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Mathematics Subject Classification: Primary: 34B15, 34B18.

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