Pointwise estimates and Liouville theorems for positive supersolutions of (p, q)-Laplace equations

Abstract

We study positive supersolutions of quasilinear elliptic problems driven by the (p, q)-Laplace operator which arise in models with nonstandard and unbalanced growth. Our approach is based on an explicit construction of radial solutions and torsion functions that serve as comparison functions. This gives a fundamental pointwise estimate for positive supersolutions, reducing the analysis to an integral condition involving a weight and a nonlinearity evaluated along a suitable radial infimum profile. As an application, we derive explicit upper bounds for the extremal parameter associated with nonlinear Dirichlet problems in bounded domains. In exterior domains, we establish optimal Liouville-type nonexistence results for (p, q)-problems. The method extends naturally to multi-phase operators, providing a unified framework for pointwise estimates and Liouville theorems in problems with nonstandard growth.