Quasilinear and Hessian Lane-Emden Type Systems with Measure Data

Abstract

We study nonlinear systems of the form \(-{\Delta }_{p}u=v^{q_{1}}+\mu , -{\Delta }_{p}v=u^{q_{2}}+\eta \) and \(F_{k}[-u]=v^{s_{1}}+\mu , F_{k}[-v]=u^{s_{2}}+\eta \) in a bounded domain Ω or in \(\mathbb {R}^{N}\) where μ and η are nonnegative Radon measures, Δp and Fk are respectively the p-Laplacian and the k-Hessian operators and q1, q2, s1 and s2 positive numbers. We give necessary and sufficient conditions for existence expressed in terms of Riesz or Bessel capacities.