## Abstract and Applied Analysis

### On the $A$-Laplacian

Noureddine Aïssaoui

#### Abstract

We prove, for Orlicz spaces $\mathbf{L}_{A}(\mathbb{R}^N)$ such that $A$ satisfies the $\Delta _2$ condition, the nonresolvability of the $A$-Laplacian equation $\Delta _{A}u+h=0$ on $\mathbb{R}^N$, where $\int h\neq 0$, if $\mathbb{R}^N$ is $A$-parabolic. For a large class of Orlicz spaces including Lebesgue spaces $\mathbf{L}^p$ ($p>1$), we also prove that the same equation, with any bounded measurable function $h$ with compact support, has a solution with gradient in $\mathbf{L}_{A}(\mathbb{R}^N)$ if $\mathbb{R}^N$ is $A$-hyperbolic.

#### Article information

Source
Abstr. Appl. Anal., Volume 2003, Number 13 (2003), 743-755.

Dates
First available in Project Euclid: 28 July 2003

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1059416442

Digital Object Identifier
doi:10.1155/S1085337503303069

Mathematical Reviews number (MathSciNet)
MR1996921

Zentralblatt MATH identifier
1080.46511

#### Citation

Aïssaoui, Noureddine. On the $A$-Laplacian. Abstr. Appl. Anal. 2003 (2003), no. 13, 743--755. doi:10.1155/S1085337503303069. https://projecteuclid.org/euclid.aaa/1059416442