| Abstract: |
| In this talk we present some new results in bifurcation theory for analytic nonlinearities obtained by J. L\'{o}pez-G\'{o}mez and the speaker. In essence, we use a number of tools from analytic geometry to determine the sharp local structure of the solution set of analytic nonlinear equations between Banach spaces. Moreover, in the context of global bifurcation theory, we generalize N. Dancer's bifurcation theorem to cover the general degenerate case, i.e., when the algebraic multiplicity of the eigenvalue is arbitrary and not necessarily equal to one. This is a breakthrough because in the case where the algebraic multiplicity is not one, the celebrated Crandall-Rabinowitz Theorem does not apply.
We conclude the talk by applying these results to certain types of semi-linear elliptic equations, describing the exact topological behavior of the set of positive and negative solutions. |
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