Liouville property in dimension five for Giaquinta’s density

Determine whether the Liouville property holds in dimension five for entire bounded solutions of the Euler–Lagrange equation associated with Giaquinta’s density f(P)=\sum_{i=1}^{n-1}P_i^2+\tfrac12 P_n^4, namely whether every bounded C^2 solution on \mathbb{R}^5 is constant.

Background

The paper proves that for separable convex densities satisfying the stated anisotropic growth assumptions, bounded entire solutions are constant when n\leq 2+\min_i p_i. For Giaquinta’s example, p_1=\cdots=p_{n-1}=2 and p_n=4, so the theorem establishes the Liouville property only for dimensions n\leq 4.

The authors explicitly identify dimension five as unresolved for the particular density underlying Giaquinta’s example. This is the only dimension immediately beyond the range covered by their theorem and is therefore a concrete unresolved case rather than a general future-work aspiration.