The $m$-Laplace equation with a gradient term II: classification in the second critical case

Abstract

Let $1<m<n$ and $0<q<m-1$. We classify positive, locally bounded weak solutions of \[-Δ_m u=u^p|Du|^q\quad\text{in }\mathbb R^n,\qquad p=\frac{m-q}{n-m}\left(n+\frac{q}{m-1-q}\right)-1.\] Every solution is constant or belongs to an explicit family of radial solutions, up to translation and scaling. No global bound, decay, or finite-energy assumption is required. The proof is based on a scalar differential identity and a maximum principle that remains valid at critical points. These lead to a sharp gradient bound, classification of normalized solutions, and a first-contact argument for arbitrary entire solutions. Local regularity and the behavior at critical points are treated throughout in the weak solution class; global $C^2$ regularity is not assumed.