Schrödinger operators involving singular potentials and measure data

Abstract

We study the existence of solutions of the Dirichlet problem for the Schrödinger operator with measure data{−Δu+Vu=μin Ω,u=0on ∂Ω. We characterize the finite measures μ for which this problem has a solution for every nonnegative potential V in the Lebesgue space Lp(Ω) with 1≤p≤N2. The full answer can be expressed in terms of the W2,p capacity for p>1, and the W1,2 (or Newtonian) capacity for p=1. We then prove the existence of a solution of the problem above when V belongs to the real Hardy space H1(Ω) and μ is diffuse with respect to the W2,1 capacity.