Liouville theorems for elliptic problems in variable exponent spaces

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

“…The conditions for the Liouville-type results for equations yielded in variable exponent settings have not yet been studied systematically in the literature. To our best knowledge, there exist five papers which are in variable settings, see [1,5,17,23,40], and only in [1,5] we find the inequality in general form with operators satisfying some structural conditions with respect to p(x)-growth.…”

“…Our result gives worse estimate on the growth. The best exponent that we can obtain is 1 − n, so this method is not sharp in general, see Remarks 12 and 13 in [17].…”

“…There are some results in the literature proved with the use of Caccioppoli-type inequality for weak subsolutions or supersolutions to PDEs. We would like to mention Liouville theorems [1,17], Harnack inequalities [2,4,25,27,39] and their further consequences-maximum and comparison principles [20,22]. The already mentioned applications are important in the regularity theory as some versions of the Harnack inequality imply that solutions are locally Hölder continuous [3].…”

“…The conditions sufficient to prove that solutions to certain problems are constant functions are called nonexistence results, i.e., nonexistence of nontrivial solutions, or the Liouville-type results. Despite multiple results of this type for various problems corresponding to (1) with constant (but not necessarilly power-type) growth [7,[10][11][12][13]21,29,38], to our best knowlegde, the variable exponent versions are considered only in [1,5,17,23,40]. Let us point out that four of them appeared in recent two years.…”

“…There, the inequalities related to our estimates are obtained as the consequence of inequalities p (u α ) 0 with a certain constant α > 0 and the inequality − p u for some locally inegrable function . Our approach to Caccioppoli-type inequalities in variable exponent setting is new and according to our best knowledge has appeared so far only in [17]. There, the Caccioppoli inequality is applied in proving a Liouville-type theorem for PDI of the form…”