The Liouville-type theorem for problems with nonstandard growth derived by Caccioppoli-type estimate (Q2173230)

The author considers the nonnegative solution \(u\) to the partial differential inequality \[ -\operatorname{div}{\mathcal A}(x, u, \nabla u) \geq {\mathcal B}(x, u, \nabla u) \] in \(\Omega \subset {\mathbb R}^n\) with \( {\mathcal A}\) and \({\mathcal B}\) differential operators with \(p(x)\)-type growth. The author first proves a Caccioppoli-type inequality for the solution \(u\) and then, as a consequence, a nice Liouville-type theorem under a suitable integral condition. With respect to the known literature, the author reduces the assumptions on the operators \({\mathcal A}\) and \( {\mathcal B}\) and does not restrict the range of \(p(x)\) by the space dimension \(n\), covering, in this way, a quite general family of problems.