Sixth Mississippi State Conference on Differential Equations and Computational Simulations.
Electron. J. Diff. Eqns., Conference 15 (2007), pp. 239-249.

On the exact multiplicity of solutions for boundary-value problems via computing the direction of bifurcations

Joaquin Rivera, Yi Li

Abstract:
We consider positive solutions of the Dirichlet problem
$$\displaylines{ u''(x)+\lambda f(u(x))=0\quad\hbox{in }(-1,1), \cr u(-1)=u(1)=0. }$$
depending on a positive parameter $\lambda $. We use two formulas derived in [18] to compute all solutions $u$ where a turn may occur and to compute the direction of the turn. As an application, we consider quintic a polynomial $f(u)$ with positive and distinct roots. For such quintic polynomials we conjecture the exact mutiplicity structure of positive solutions and present computer assisted proofs of such exact bifurcation diagrams for various distributions of the real roots. The limiting behavior of the solutions on these bifurcation branches as $\lambda \to \infty $ and their stabilities are also investigated.

Published February 28, 2007.
Math Subject Classifications: 34B15.
Key Words: Bifurcation points; direction of the turn; multiplicity of solutions.

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Joaquin Rivera
Department of Mathematics, University of Iowa
Iowa City, Iowa 52242, USA
email: rvera@math.uiowa.edu
Yi Li
Department of Mathematics, University of Iowa
Iowa City, Iowa 52242, USA.
Hunan Normal University
Changsha 410081, Hunan, China
email: yi-li@uiowa.edu

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