Liouville theorems for elliptic problems in variable exponent spaces

SYLWIA DUDEK; IWONA SKRZYPCZAK

doi:10.3934/cpaa.2017026

Article Contents

2017, Volume 16, Issue 2: 513-532. Doi: 10.3934/cpaa.2017026

This issue Previous Article Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity Next Article Asymptotic behavior of solutions to a nonlinear plate equation with memory

Liouville theorems for elliptic problems in variable exponent spaces

SYLWIA DUDEK^1, and
IWONA SKRZYPCZAK^2,

1.
Institute of Mathematics, Krakow University of Technology, ul. Warszawska 24, 31-155 Krakow, Poland
2.
Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland

Author Bio: E-mail address: sylwia.dudek@pk.edu.pl; E-mail address: iskrzypczak@mimuw.edu.pl

Received: April 30, 2016

Revised: October 31, 2016

Published: August 2017

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Abstract

We investigate nonexistence of nonnegative solutions to a partial differential inequality involving the p(x){Laplacian of the form

$- {\Delta _{p(x)}}u \geqslant \Phi (x,u(x),\nabla u(x))$

in ${\mathbb{R}^n}$, as well as in outer domain $\Omega \subseteq {\mathbb{R}^n}$, where Φ(x; u; ∇u) is a locally integrable Carathéodory’s function. We assume that Φ(x; u; ∇u) ≥ 0 or compatible with p and u. Growth conditions on u and p lead to Liouville-type results for u.

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Mathematics Subject Classification: 26D10, 35J60, 35J91.

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