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

Abstract

Let u be a nonnegative solution to the PDI − divA(x, u, ∇u) B(x, u, ∇u) in , where A and B are differential operators with p(x)-type growth. As a consequence of the Caccioppoli-type inequality for the solution u, we obtain the Liouville-type theorem under some integral condition. We simplify the assumptions on functions A and B, and we do not restrict the range of p(x) by the dimension n, therefore we can cover quite general family of problems.

Small weights in Caccioppoli's inequality and applications to Liouville-type theorems for non-standard problems

“…Contributions in the case of linear or anisotropic growth are quite rarely found. We just mention the papers [3], [4], [5], [6] and the references quoted therein.…”