Bifurcation for Lipschitz Operators With An Application To Elasticity.
Technical summary rept.,
WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER
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The authors consider operator equations of the form A sub 0 - lambda sub 0B sub 0u Nlambda, u 1 where A sub 0, B sub 0 are linear operators between real Banach spaces and Nlambda, u is a nonlinear operator with the property that Nlambda, 0 0 for all real lambda. Assuming that lambda sub 0, a specific value of lambda, is an isolated eigenvalue of A sub 0 - lambda B sub 0 of multiplicity m the authors study the phenomenon of bifurcation for equation 1, where it is merely assumed that Nlambda, u is Lipschitz continuous in u near u 0 with a small Lipschitz constant. It is shown that when 1 has a variational structure, for each suitable normalization of u, two non-zero solutions lambda, u occur near lambda sub 0, 0 m pairs occur if N is odd in u. Further results concern the existence of branches of solutions when m is odd and the asymptotic behavior of solutions in terms of the size of the Lipschitz constant. The motivation for the study and the main application of the results concerns buckling of a von Karman plate resting on a foundation.
- Theoretical Mathematics