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Dudek, Sylwia

The Liouville-type theorem for problems with nonstandard growth derived by Caccioppoli-type estimate. (English) Zbl 1439.35108

Monatsh. Math. 192, No. 1, 75-91 (2020).
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.
Reviewer: Vincenzo Vespri (Firenze)

Cited in 1 Document

MSC:

35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
26D10 Inequalities involving derivatives and differential and integral operators
35J60 Nonlinear elliptic equations
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35R45 Partial differential inequalities and systems of partial differential inequalities

Keywords:

Caccioppoli inequality; variable exponent Lebesgue space
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References:

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