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

Abstract

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 \(p-1<α<q^*-1\), where \(q^*=nq/(n-q)\). 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.