A variational approach to a class of nonlinear eigenvalue problems.

Abstract:

Let f be a real-valued differentiable function defined on the real reflexive Banach space X. The problem of minimizing f over a subset of X is investigated under the following mild monotonicity assumption on the derivative $f’$ of f: if $\{ {u_n}\}$ is a sequence in X converging weakly to some $u \in X$, then $\lim \sup (f’{u_n},{u_n} - u) \geqq 0$ . The eigenvalue problem $f’u = \lambda g’u$ for some $\lambda \in {R^1}$, with $g’$ being the derivative of a further function g, is then reduced to that first question.