Variational and non-variational eigenvalues of the p-Laplacian

It is well known that all the eigenvalues of the linear eigenvalue problem? u = (q - ? r) u, in  O ? R^N, can (under appropriate conditions on q, r and O) be characterized by minimax principles, but it has been a long-standing question whether that remains true for analogous equations involving the p-Laplacian ?_p. It will be shown that there are corresponding nonlinear eigenvalue problems?_p u = (q - ? r) | u |^{p - 1} sgn u, in  O ? R^N, with 1 < p ? 2 and q, r ? C¹ (over(O, -)), r > 0 on over(O, -), for which not all eigenvalues are of variational type. As far as we know, this is the first observation of such a phenomenon, and examples will be given for one- and higher-dimensional equations. The question of exactly which eigenvalues are variational is also discussed when N = 1. © 2007 Elsevier Inc. All rights reserved.