The basic higher order commutator theorem is formulated for the real interpolation methods associated with the quasi-power parameters, that is, the function spaces on which Hardy inequalities are valid. This theorem unifies and extends various results given by Cwikel, Jawerth, Milman, Rochberg, and others, and incorporates some results of Kalton to the context of commutator estimates for the real interpolation methods.

We study quasilinear hemivariational inequalities involving the $p$-Laplacian. We prove two existence theorems. In the first, we allow “crossing” of the principal eigenvalue by the generalized potential, while in the second, we incorporate problems at resonance. Our approach is based on the nonsmooth critical point theory for locally Lipschitz energy functionals.

We prove the simplicity and isolation of the first eigenvalue for the problem ${\Delta }_{p}u={\left|u\right|}^{p-2}u$ in a bounded smooth domain $\Omega \subset {ℝ}^N$, with a nonlinear boundary condition given by ${\left|\nabla u\right|}^{p-2}\partial u/\partial v=\lambda {\left|u\right|}^{p-2}u$ on the boundary of the domain.