Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 107 / 3 / c / Solution

Nonnegativity lets us dispense with a sign restriction on $c$. Set $c_0=\min (c,0)$ and $L_0=a^{ij}{D}_{ij}+b^i{D}_i+c_0$. ThenThe strong minimum principle for elliptic operators forces $u$ to vanish identically if it has an interior zero. Otherwise it is strictly positive:Connectedness is part of the usual domain hypothesis; on a disconnected open set this conclusion applies separately to each component.