Schroedinger operators involving singular potentials and measure data

Augusto C. Ponce, Nicolas Wilmet

Published 2017-05-10Version 1

We study the existence of solutions of the Dirichlet problem for the Schroedinger operator with measure data $$ \left\{ {2} -\Delta u + Vu &= \mu and \quad \text{in \(\Omega\)}, u &= 0 && \quad \text{on \(\partial\Omega\)}. \right. $$ We characterize the finite measures $\mu$ for which this problem has a solution for every nonnegative potential $V$ in the Lebesgue space $L^p(\Omega)$ with $1 \le p \le N/2$. The full answer can be expressed in terms of the $W^{2,p}$ capacity for $p > 1$, and the $W^{1,2}$ (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 $H^1(\Omega)$ and $\mu$ is diffuse with respect to the $W^{2,1}$ capacity.