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

Abstract

Let $1<m<n$, $q>0$, and $p\in\mathbb R$. We study positive $C^1_{\mathrm{loc}}$ weak solutions of \[-\Delta_m u=u^p|Du|^q\qquad\text{in }\mathbb R^n.\] If $0<q<m-1$, we prove that every such solution is constant provided \[p+q-m+1<\frac{(m-1)(m-q)^2}{(n-m)(m-1-q)}.\] If $q\geqslant m-1$, 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 $\{|Du|=0\}$. When $m=2$, our result answers the natural Liouville problem left open by Bidaut-V\'eron, Garc\'ia-Huidobro and V\'eron 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-V\'eron, while the case $q>m$ follows from the theorem of Lu and Zhu.