Liouville-Type Result for Quasilinear Elliptic Problems With Variable Exponent

Abstract

In this paper we discuss several aspects of relations between the Keller-Osserman property and the validity of the Liouville-type theorem for quasilinear elliptic equation with variable exponent that include non-homogeneous p(.)-Laplace equations for p measurable function.

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

“…The Liouville-type theorems for problems involving such A are proved in e.g. [1,5]. In this papers we find diffrent approach to ours.…”

“…imply the Liouville-type theorem for (15). The Liouville-type theorem for quasilinear elliptic equations in R n with variable exponent appears also in [5], where the following equation is investigated…”

“…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.…”

“…|ζ | 2 for some κ > 0 and with a measurable and bounded function d 2 : → R + (such operators B are not allowed in [1,5]).…”

“…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.…”