Giaquinta's example revisited: the Liouville property for a class of nonuniformly elliptic equations in low dimensions

Abstract

We consider entire solutions $u$: ${\mathbb{R}}^n\to \mathbb{R}$, $n\ge 2$, of the equation $\mathrm{div}(\nabla f(\nabla u))=0$ with convex density $f$: ${\mathbb{R}}^n\to \mathbb{R}$ given for example by $f(\nabla u)={\sum }_{i=1}^n|{\partial }_{i}u{|}^{p_i}$ with exponents $p_i\ge 2$ including Giaquinta's example for the choice $p_1=\cdots =p_{n-1}=2$, $p_n=4$. If $n\le 2+\min \{p_i\}$ holds, then we have the Liouville property: the boundedness of $u$ implies its constancy.