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

Abstract

We consider entire solutions $u$: $\mathbb{R}^n \rightarrow \mathbb{R}$, $n \ge 2$, of the equation $\operatorname{div}(\nabla f(\nabla u)) = 0$ with convex density $f$: $\mathbb{R}^n \rightarrow \mathbb{R}$ 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 $p_1 = \dots = p_{n-1} = 2$, $p_n = 4$. If $n \le 2 + \min\{p_i\}$ holds, then we have the Liouville property: the boundedness of $u$ implies its constancy.