Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Global analysis of two-parameter elliptic eigenvalue problems

Authors: H.-O. Peitgen and K. Schmitt
Journal: Trans. Amer. Math. Soc. 283 (1984), 57-95
MSC: Primary 35B32; Secondary 34B15, 35J65, 47H15, 58E07
MathSciNet review: 735409
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Abstract: We consider the nonlinear boundary value problem $ ({\ast})Lu + \lambda f(u) = 0$, $ x \in \Omega ,\,u = \sigma \phi ,\,x \in \partial \Omega $, where $ L$ is a second order elliptic operator and $ \lambda $ and $ \sigma $ are parameters. We analyze global properties of solution continua of these problems as $ \lambda $ and $ \sigma $ vary. This is done by investigating particular sections, and special interest is devoted to questions of how solutions of the $ \sigma = 0$ problem are embedded in the two-parameter family of solutions of $ ({\ast})$. As a natural biproduct of these results we obtain (a) a new abstract method to analyze bifurcation from infinity, (b) an unfolding of the bifurcations from zero and from infinity, and (c) a new framework for the numerical computations, via numerical continuation techniques, of solutions by computing particular one-dimensional sections.

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Article copyright: © Copyright 1984 American Mathematical Society