The author uses a version of Gårding's inequality to obtain a (nodally oscillatory) comparison theorem which can then be used to extend his earlier non-oscillation theorems [the author, Math. Nachr. 141, 289- 297 (1989; Zbl 0682.35008)] to the equation \[ \sum_{|\alpha|,|\beta|=0}^ m(- 1)^{|\alpha|}D^ \alpha [A_{\alpha\beta}(x)D^ \beta u]=0, \] where \(x\in\Omega\subset\mathbb{R}^ n\) and \(\Omega\) is an unbounded open set. Here the principal coefficients \(A_{\alpha\beta} (|\alpha|=|\beta|=m)\) are uniformly continuous in \(\Omega\) and the other coefficients are bounded and measurable on \(\Omega\).