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Imperfect bifurcations arising from elliptic boundary value problems. (English) Zbl 1017.34041
Bifurcations for the boundary value problem \[ \Delta u+\lambda(u^p-u)=0\quad\text{ in} \Omega,\quad u>0\quad\text{ in }\Omega,\quad{\partial u\over\partial \nu}+\varepsilon u=0\quad\text{ on }\quad \partial\Omega, \] are studied. Here, \(\Omega\) is the unit ball centered at the origin, \(p>1\), \(\varepsilon, \lambda\) are parameters. In the case of \(\varepsilon=0\) (Neumann problem), bifurcations from \(u\equiv 1\) has been studied in detail by many researches and many interesting results are known. The paper concentrates on the question what happens if \(\varepsilon\) is slightly perturbed from zero. In this case, \(u\equiv 1\) is not a solution any more which affects the topology of the bifurcations branches and leads to so-called imperfect bifurcations in turn; see e.g., M. Golubitsky and D. G. Schaeffer [Singularities and groups in bifurcation theory. Volume I. New York etc.: Springer-Verlag (1985; Zbl 0607.35004)].
Reviewer: Petr Girg (Plzen)

MSC:
34C23 Bifurcation theory for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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