A compact support principle for singular elliptic differential inequalities with a gradient term

In this paper, a compact support principle is established for the elliptic differential inequality Δu+|∇u|p≥K(x)f(u),u≥0, for |x||x| large, where p≥1p≥1, K(x)≥0K(x)≥0, and ff satisfies conditions (F1)–(F3) below. The main feature of this note is the presence of the gradient term |∇u|p|∇u|p and the singular coefficient function K(x)K(x). The result is optimal in some sense related to the power pp and the decaying rate of K(x)K(x) at ∞∞, and the proof is based on finding appropriate comparison functions. We also give a similar result in a bounded domain ΩΩ of RN(N≥2)RN(N≥2).