Second eigenfunctions of nonlinear eigenvalue problems (Q1910061)

The author considers the eigenvalue problem (1) \(g'(u) = \lambda f' (u)\) where \(f\) and \(g\) are Fréchet differentiable functionals on a Hilbert space \(H\). A particular case of (1) is a linear equation \(Au = \lambda u\) where \(A\) is a weakly continuous selfadjoint linear operator on \(H\). The problem (1) is viewed as the critical point equation for \(g\) on the set \(\{u \in H : f(u) \leq t\}\). The constrained minimax problem of the so-called ``mountain pass type'' provides solutions of (1) which may be different from solutions obtained by evaluating Lagrange multipliers based on the function of critical values \(\gamma (t) = \inf_{f (u) \leq t} g(u)\). In the particular case of the linear equation mentioned before, these solutions happen to be the second eigenvalue and the vectors of the corresponding eigenspace, whereas the latter method gives the first eigenvalue and the vectors of the associated eigenspace. The main result of the paper is that, under certain quite restrictive conditions on \(f\) and \(g\), the critical value \(\sigma (t)\) associated with (1) has right and left hand derivatives which are the eigenvalues of (1). The eigenvalues \(\sigma_\pm' (t)\) are distinguished among other eigenvalues by certain minimax relations. Finally, the results are applied to semilinear elliptic problems.