Comparison principles for p-Laplace equations with lower order terms

We prove comparison principles for quasilinear elliptic equations whose simplest model is $$\begin{aligned} \lambda u -\Delta _p u + H(x,Du)=0 \quad x\in \Omega , \end{aligned}$$ where \(\Delta _p u = \text { div }(|Du|^{p-2} Du)\) is the p-Laplace operator with \(p> 2\), \(\lambda \ge 0\), \(H(x,\xi ):\Omega \times \mathbb {R}^{N}\rightarrow \mathbb {R}\) is a Caratheodory function and \(\Omega \subset \mathbb {R}^{N}\) is a bounded domain, \(N\ge 2\). We collect several comparison results for weak sub- and super-solutions under different setting of assumptions and with possibly different methods. A strong comparison result is also proved for more regular solutions.