Liouville Theorem for $(p,q)$-Laplace Equations

We employ the vector field method to establish a Liouville-type theorem for a class of $(p,q)$-Laplace equations in the Euclidean space ${\mathbb{R}}^n$. By modifying the exponents in the differential identity, we prove nonexistence in the subcritical range $p-1<\alpha <q^{\ast }-1$, where $q^{\ast }=nq\mathrm{/}(n-q)$. The approach relies on constructing a suitable differential identity, carrying out precise integral estimates with cutoff functions, and combining sign control and decay of the cutoff errors.