Variational and non-variational eigenvalues of the p-Laplacian

It is well known that all the eigenvalues of the linear eigenvalue problemΔu=(q−λr)u,in Ω⊂RN, can (under appropriate conditions on q, r and Ω) be characterized by minimax principles, but it has been a long-standing question whether that remains true for analogous equations involving the p  -Laplacian ΔpΔp. It will be shown that there are corresponding nonlinear eigenvalue problemsΔpu=(q−λr)|u|p−1sgnu,in Ω⊂RN, with 10r>0 on Ω¯, 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=1N=1.