Summary: This paper treats the quasilinear elliptic inequality \[ \text{div}(|Du|^{m-2}Du)\geq p(x)u^\sigma,\quad x\in\mathbb{R}^N, \] where \(N\geq 2\), \(m>1\), \(\sigma>m-1\), and \(p:\mathbb{R}^N\to(0,\infty)\) is continuous. Sufficient conditions are given for this inequality to have no positive entire solutions. When \(p\) has radial symmetry, the existence of positive entire solutions can be characterized by our results and some known results.