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On bifurcation from infinity for potential operators. (English) Zbl 0736.58014

The authors consider the equation (1) \(Lu+G(u)=\lambda u\) where \(L\) is a bounded linear selfadjoint operator on the Hilbert space \(X\) and \(G\in C(\Omega,X)\) is of potential type. They are concerned with solutions of equation (1) which bifurcate from infinity: more precisely \((\mu,\infty)\) is a bifurcation point from infinity if, for any \(\varepsilon>0\) and \(M>0\), there exists a solution \((\lambda,u)\) such that \(| \lambda- \mu|<\varepsilon\) and \(\| u\|>M\). The aim of this paper is to prove that under some conditions the parameter values of \(\lambda\) of solutions bifurcating from \((\mu,\infty)\) cover at a least one-sided neighborhood of \(\mu\) provided there are no solutions of equation \((1)_ \mu\) tending towards infinity, where \((1)_ \mu\) is equation (1) with \(\lambda=\mu\).
Reviewer: M.A.Vivaldi (Roma)

MSC:

58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
35B32 Bifurcations in context of PDEs
47J05 Equations involving nonlinear operators (general)
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