Quasilinear Operators with Natural Growth Terms

We will describe some joint work with V. G. Maz’ya and I. E. Verbitsky, concerning homogeneous quasilinear differential operators. The model operator under consideration is:

\[ L(u) = - \Delta_p u - \sigma |u|^{p-2} u. \]

Here $\Delta_p$ is the p-Laplacian operator and $\sigma$ is a signed measure, or more generally a distribution. We will discuss an approach to studying the operator L under only necessary conditions on $\sigma$, along with applications to the characterisation of certain Sobolev inequalities with indefinite weight. Many of the results discussed are new in the classical case p = 2, when the operator L reduces to the time independent Schrödinger operator.