A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient

TL;DR

Addresses Liouville-type nonexistence for positive bounded solutions of the elliptic equation $-\Delta u = u^q |\nabla u|^p$ in $\mathbb{R}^n$ for $p\ge 2$ and $q>0$, proving constancy of such solutions. The approach combines the monotone nonincreasing behavior of spherical averages of superharmonic functions with a local Bernstein argument to obtain a gradient bound, and uses a subharmonicity argument on $z=(u-\ell)^s$ with $\ell=\lim_{R\to\infty}\bar u(R)$ to conclude. The work clarifies the Liouville property, showing nonconstant bounded supersolutions exist when $(n-2)q+(n-1)p>n$, and extends the method to certain elliptic systems under growth conditions. It situates the results relative to the Lane-Emden equation and prior works in the range $0<p<2$, offering new insights for the superquadratic-gradient regime.

Abstract

We consider the elliptic equation $-Δu = u^q|\nabla u|^p$ in $\mathbb R^n$ for any $p\ge 2$ and $q>0$. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in~\cite{BVGHV}, where the authors consider the case $0<p<2$. Some extensions to elliptic systems are also given.

Theorems & Definitions (10)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Lemma 2.1
• Lemma 2.2
• proof : Proof of Lemma \ref{['lembound']}
• Remark 2.1
• proof : Proof of Theorem \ref{['mainthm']}
• proof : Proof of Theorem \ref{['mainthm2']}