Classification of bifurcation diagrams of a $P$-Laplaciannonpositone problem

Po-Chun Huang; Shin-Hwa Wang; Tzung-Shin Yeh

doi:10.3934/cpaa.2013.12.2297

Article Contents

2013, Volume 12, Issue 5: 2297-2318. Doi: 10.3934/cpaa.2013.12.2297

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Classification of bifurcation diagrams of a $P$-Laplacian nonpositone problem

1.
Department of Mathematics, National Tsing Hua University, Hsinchu 300
2.
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300
3.
Department of Applied Mathematics, National University of Tainan, Tainan 700

Received: January 31, 2012

Revised: March 31, 2012

Published: August 2013

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Abstract

We study the bifurcation diagrams of positive solutions of the $p$-Laplacian Dirichlet problem \begin{eqnarray*} (\varphi_p(u'(x)))'+f_\lambda(u(x))=0, -1 < x < 1, \\ u(-1)=u(1)=0, \end{eqnarray*} where $\varphi_p(y)=|y|^{p-2}y$, $(\varphi_p(u'))'$ is the one-dimensional $p$-Laplacian, $p>1$, the nonlinearity $f_\lambda(u)=\lambda g(u)-h(u),$ $g,h\in C[0,\infty)\cap C^2(0,\infty )$, and $\lambda >0$ is a bifurcation parameter. Under certain hypotheses on functions $g$ and $h$, we give a complete classification of bifurcation diagrams. We prove that, on the $(\lambda, |u|_\infty)$-plane, each bifurcation diagram consists of exactly one curve which has exactly one turning point where the curve turns to the right. Hence we are able to determine the exact multiplicity of positive solutions for each $\lambda >0.$ In addition, we show the evolution phenomena of bifurcation diagrams of polynomial nonlinearities with positive coefficients.

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Mathematics Subject Classification: 34B15, 34B18.

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