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

Abstract

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