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Bifurcation of gradient mappings possessing the Palais-Smale condition. (English) Zbl 1233.37028

It is showed that the compactness inherent in the Palais-Smale condition is an adequate substitute for the requirement of complete continuity of the perturbation. The authors use the technique of Krasnosel’skii and do not implement Lyapunov-Schmidt reduction. Specific estimates for the size of the solution are obtained and the local estimates of Chiappinelli are extended. By adopting an alternative approach to identify the bifurcation branch, the point at which compactness is required is identified and weakened to a local Palais-Smale condition. Some applications of bifurcation theorems are presented.

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
37D15 Morse-Smale systems
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
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References:

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