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The effects of nonlinear perturbation terms on comparison principles for the $p$ $p$ -Laplacian

Ahmed Mohammed

https://orcid.org/0000-0001-5727-3614

Department of Mathematical Sciences, Ball State University, Muncie, Indiana 47306, USA

E-mail Address: amohammed@bsu.edu

Corresponding author.

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and

Antonio Vitolo

Dipartimento di Matematica, Università di Salerno, Fisciano, Salerno 84084, Italy

E-mail Address: vitolo@unisa.it

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https://doi.org/10.1142/S166436072450005XCited by:3 (Source: Crossref)

Abstract

In this paper, we investigate various comparison principles for quasilinear elliptic equations of $p$ $p$ -Laplace type with lower-order terms that depend on the solution and its gradient. More specifically, we study comparison principles for equations of the following form:

- Δ p u + H (u, D u) = 0, x \in Ω, - Δ_{p} u + H (u, D u) = 0, x \in Ω, <math display="block" altimg="eq-00003.gif"><mrow><mo stretchy="false" class="MathClass-bin">-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mi>u</mi><mo stretchy="false" class="MathClass-bin">+</mo><mi>H</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>u</mi><mo class="MathClass-punc">,</mo><mi>D</mi><mi>u</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">=</mo><mn>0</mn><mo class="MathClass-punc">,</mo><mspace width="1em" class="quad"></mspace><mi>x</mi><mo class="MathClass-rel">\in</mo><mi mathvariant="normal">Ω</mi><mo class="MathClass-punc">,</mo></mrow></math>

where

Δ_{p} u : = div (∣ ∣ D u {∣ ∣}^{p - 2} D u)

$Δ_{p} u : = div (| D u |^{p - 2} D u)$ is the

p

$p$ -Laplace operator with

p > 1

$p > 1$ , and

H

$H$ is a continuous function that satisfies a structure condition. Many of these results lead to comparison principles for the model equations

Δ p u = f (u) + g (u) | D u | q, x \in Ω, Δ_{p} u = f (u) + g (u) | D u |^{q}, x \in Ω, <math display="block" altimg="eq-00008.gif"><mrow><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mi>u</mi><mo class="MathClass-rel">=</mo><mi>f</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>u</mi><mo class="MathClass-close" stretchy="false">)</mo><mo stretchy="false" class="MathClass-bin">+</mo><mi>g</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>u</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">|</mo><mi>D</mi><mi>u</mi><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mi>q</mi></mrow></msup><mo class="MathClass-punc">,</mo><mspace width="1em" class="quad"></mspace><mi>x</mi><mo class="MathClass-rel">\in</mo><mi mathvariant="normal">Ω</mi><mo class="MathClass-punc">,</mo></mrow></math>

where

$f, g \in C^{0} (ℝ, ℝ)$ are non-decreasing and

$q > 0$ . Our results either improve or complement those that appear in the literature.

Communicated by Vicentiu Radulescu

Keywords:

AMSC: 35J62, 35B50, 35B51

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Vol. 14, No. 02

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History

Received 13 November 2023

Revised 4 May 2024

Accepted 6 May 2024

Published: 30 May 2024

Information

This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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The effects of nonlinear perturbation terms on comparison principles for the pp<math altimg="eq-00001.gif" display="inline"><mi>p</mi></math>-Laplacian

Abstract

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The effects of nonlinear perturbation terms on comparison principles for the $p$ $p$ -Laplacian