On the existence of weak solutions of semilinear elliptic equations and systems with Hardy potentials

Abstract

© 2018 Elsevier Inc.Let Ω⊂RN (N≥3) be a bounded C2 domain and δ(x)=dist(x,∂Ω). Put Lμ=Δ+[Formula presented] with μ>0. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to −Lμu=up+τin Ω,u=νon ∂Ω where μ>0, p>0, τ and ν are measures on Ω and ∂Ω respectively. We then establish existence results for the system {−Lμu=ϵvp+τin Ω,−Lμv=ϵup˜+τ˜in Ω,u=ν,v=ν˜on ∂Ω where ϵ=±1, p>0, p˜>0, τ and τ˜ are measures on Ω ν and ν˜ are measures on ∂Ω. We also deal with elliptic systems where the nonlinearities are more general.