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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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