Liouville comparison principles for solutions of semilinear elliptic second-order partial differential inequalities

The purpose of this work is to obtain new Liouville comparison principles for entire weak solutions of semilinear elliptic second-order partial differential inequalities of the forms Lu + vertical bar u vertical bar(q-1) u <= Lv + vertical bar v vertical bar(q-1)v and -Lu + vertical bar u vertical bar(q-1) u <= -Lv + vertical bar v vertical bar(q-1)v on R-n, where n >= 2, q>0 and L is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergence form, in terms of the behaviour of the coefficients of the operator L at infinity in a function space directly associated with L.