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 $$f(\nabla u) = \sum_{i=1}ⁿ |\partial_i u|<sup>{p_i}$$ 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.