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# International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

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Volume 19, Issue 2

# Variational Approaches to P(X)-Laplacian-Like Problems with Neumann Condition Originated from a Capillary Phenomena

Shapour Heidarkhani
/ Ghasem A. Afrouzi
Published Online: 2018-02-08 | DOI: https://doi.org/10.1515/ijnsns-2017-0114

## Abstract

This article presents several sufficient conditions for the existence of at least one weak solution and infinitely many weak solutions for the following Neumann problem, originated from a capillary phenomena, $\left\{\begin{array}{l}-\mathrm{d}\mathrm{i}\mathrm{v}\left(\left(1+\frac{|\mathrm{\nabla }u{|}^{p\left(x\right)}}{\sqrt{1+|\mathrm{\nabla }u{|}^{2p\left(x\right)}}}\right)|\mathrm{\nabla }u{|}^{p\left(x\right)-2}\mathrm{\nabla }u\right)+\alpha \left(x\right)|u{|}^{p\left(x\right)-2}u\\ =\lambda f\left(x,u\right)\text{in}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\Omega },\\ \frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }=0\text{on}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\partial }\mathrm{\Omega }\end{array}$

where $\mathrm{\Omega }\subset {\mathbb{R}}^{N}$ $\left(N\ge 2\right)$ is a bounded domain with boundary of class ${C}^{1},$ $\nu$ is the outer unit normal to $\mathrm{\partial }\mathrm{\Omega },$ $\lambda >0$, $\alpha \in {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right),$ $f:\mathrm{\Omega }×\mathbb{R}\to \mathbb{R}$ is an ${L}^{1}$-Carathéodory function and $p\in {C}^{0}\left(\stackrel{‾}{\mathrm{\Omega }}\right)$. Our technical approach is based on variational methods and we use a more precise version of Ricceri’s Variational Principle due to Bonanno and Molica Bisci. Some recent results are extended and improved. Some examples are presented to illustrate the application of our main results.

MSC 2010: Primary 35D05; Secondary 35J60

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Accepted: 2018-01-15

Published Online: 2018-02-08

Published in Print: 2018-04-25

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 2, Pages 189–203, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339,

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