The $m$-Laplace equation with a gradient term I: Liouville theorem in the second subcritical case

Let $1<m<n$, $q>0$, and $p\in \mathbb{R}$. We study positive $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ weak solutions of $$-{\mathrm{\Delta }}_{m}u=u^p\mathrm{\mid }Du{\mathrm{\mid }}^q\text{in }{\mathbb{R}}^n\mathrm{.}$$ If $0<q<m-1$, we prove that every such solution is constant provided $$p+q-m+1<\frac{(m-1)(m-q{)}^2}{(n-m)(m-1-q)}\mathrm{.}$$ If $q⩾m-1$, the same conclusion holds for every $p\in \mathbb{R}$. For $0<q<m$, the proof uses an auxiliary function and reduces the main differential estimate to a lower bound for a function of one variable. An explicit construction and a local maximum-principle argument then prove constancy without differentiating across $\{\mathrm{\mid }Du\mathrm{\mid }=0\}$. When $m=2$, our result answers the natural Liouville problem left open by Bidaut-Véron, García-Huidobro and Véron for the whole range $0<q<1$. The case $q=m$ is handled by an increasing change of the dependent variable, in the spirit of the related work of Bidaut-Véron, while the case $q>m$ follows from the theorem of Lu and Zhu.