Consider operators $L_{ V}:=\Delta + V$ in a bounded smooth domain $D$ in $R^N$. Assume that $V\in C^1(D)$ and $V$ may blow up at the boundary at most as $1/\delta^2$ where $\delta$ denotes distance to the boundary. Assume also that $L_{ V}$ has a ground state $\Phi_V$ that satsifies an additional condition on its behavior near the boundary (see Section 3). These conditions are satisfied by a large class of potentials (see Section 6). We derive sharp, two-sided estimates of weighted integrals of positive $L_V$ harmonic functions and $L_V$ potentials. These lead to a-priori estimates of positive $L_V$ supersolutions and subsolutions assuming (in the latter case) existence of $L_V$ boundary trace.
Categories: math.AP