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

Using a variant of Caccioppoli’s inequality involving small weights, i.e. weights of the form $(1+\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2{)}^{-\alpha \mathrm{/}2}$ for some $\alpha >0$, we establish several Liouville-type theorems under general non-standard growth conditions.