Classification of bifurcation diagrams of a p-Laplacian Dirichlet problem with examples

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:369

Article

Wang, Shin-Hwa²  Yeh, Tzung-Shin¹ 

[1] Natl Univ Tainan, Dept Math Educ, Tainan 700, Taiwan

[2] Natl Tsing Hua Univ, Dept Math, Hsinchu 300, Taiwan

关键词: Bifurcation diagram;    Positive solution;    Exact multiplicity;    p-Laplacian;    Time map;   

DOI  :  10.1016/j.jmaa.2010.02.057

来源: Elsevier

【 摘 要 】

We study bifurcation diagrams of positive solutions of the p-Laplacian Dirichlet problem {(phi p(u'(x))' + f(lambda)(u(x)) = 0, -1 < x < 1, u(-1) = u(1) = 0, where phi(p)(y)= vertical bar y vertical bar(p-2) y, (phi(p)(u'))' is the one-dimensional p-Laplacian, and p > 1 and lambda > 0 are two bifurcation parameters. Assume that f(lambda)(u) = lambda(g)(u) - h(u) where g, h is an element of C[0, infinity) boolean AND C-2(0, infinity) satisfy hypotheses (H1)-(H5) presented herein. For different values p with 1 < p <= 2 and with p > 2, we give a classification of totally six different bifurcation diagrams. We prove that, on the (lambda, parallel to u parallel to(infinity))-plane, each possible bifurcation diagram consists of exactly one curve with exactly one turning point where the curve turns to the right. Hence we are able to determine the exact multiplicity of positive solutions. In addition, for 1 < p 2 and for p > 2, we give interesting examples f(lambda)(u) = lambda(ku(p-1) + u(q)) - u(r) satisfying r > q > p - 1 and k >= 0, and show complete evolution of bifurcation diagrams as evolution parameter k varies from 0 to infinity. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

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