On non-negative solutions of quasilinear elliptic inequalities

We study the solutions of the inequalities
$$ Lu\geqslant F(x,u), \qquad \mathcal Lu\geqslant F(x,u), $$
where
$$ L=\sum_{i,j=1}^n\frac\partial{\partial x_i}\biggl(a_{ij}(x)\frac\partial{\partial x_j}\biggr), \qquad \mathcal L=\sum_{i,j=1}^n a_{ij}(x)\frac{\partial^2}{\partial x_i\,\partial x_j}\,, $$
are differential operators of elliptic type and $F$ is some function.