In the paper the differential inequality ∆pu + B(x, u) ≤ 0, where ∆pu = div(||∇u|| p−2∇u), p > 1, B(x, u) ∈ C(Rn × R, R) is studied. Sufficient conditions on the function B(x, u) are established, which guarantee nonexistence of an eventually positive solution. The generalized Riccati transformation is the main tool.