Branching points for a class of variational operators
Bifurcation problems of the type \[ Lu + H(u) = \lambda u \] in a Hilbert space \(E\) are studied. Here \(L:E \to E\) is a linear symmetric Fredholm operator, \(H(u) = h'(u)\), \(h\) is a functional on \(E\), \(h(0)=h'(0)=h''(0)=0, \lambda\) is a real parameter. It is proved that for a large class of functionals, any eigenvalue \(\mu\) of odd multiplicity is a branching point in the sense that the closure \(\Sigma\) of the set of nontrivial solutions contains a connected set \(S\) such that \([\mu,0] \in S, \;S \setminus\{[\mu,0]\} \neq \emptyset \). (Let us recall that known counterexamples show that in general, \(\Sigma\) can contain only isolated points.) Moreover, under natural assumptions about the higher derivatives of \(h\) at zero, it is proved that the set \(\Pi S = \{\lambda \in R;\;[\lambda,u] \in \Sigma \text{ for some } u\}\) contains an interval around \(\mu\) or a one sided neighbourhood \(U\) of \(\mu\). Further, for any \(\mu \in U\) there are at least two nontrivial solutions, and in the case of a multiple eigenvalue at least two couples of nontrivial solutions on \(S\). Examples of boundary value problems for PDE are given.
scientific article; zbMATH DE number 1274834
scientific article