Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces Via Leray-Schauder's nonlinear alternative

Abdelaaziz Sbai; Youssef El Hadfi; Mohammed Srati; Noureddine Aboutabit

doi:10.3934/dcdss.2021015

Article Contents

2022, Volume 15, Issue 1: 213-227. Doi: 10.3934/dcdss.2021015

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$ p(x)- $

Laplacian : Application of the Nehari method

Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces Via Leray-Schauder's nonlinear alternative

1.
Laboratory LIPIM, National School of Applied Sciences Khouribga, Sultan Moulay Slimane University, Morocco
2.
Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz, Laboratory of Mathematical Analysis and Applications, Fez, Morocco

^* Corresponding author: yelhadfi@gmail.com
^* Corresponding author: yelhadfi@gmail.com

Received: August 31, 2020

Revised: December 31, 2020

Early access: January 2021

Published: January 2022

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Abstract

In this paper, we establish the existence of weak solution in Orlicz-Sobolev space for the following Kirchhoff type probelm

$ \begin{equation*} \left\{ \begin{array}{ll} -M\left( \int_{\Omega}\varPhi(|\nabla u|)dx\right) div(a(|\nabla u|)\nabla u) = f(x, u) \, in \, \, \, \, \Omega, \\ u = 0 \, \, \, \, on\, \, \, \, \, \, \, \, \, \, \partial \Omega, \end{array} \right. \end{equation*} $

where $ \Omega $ is a bounded subset in $ {\mathbb{R}}^N $, $ N\geq 1 $ with Lipschitz boundary $ \partial \Omega. $ The used technical approach is mainly based on Leray-Shauder's non linear alternative.

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Mathematics Subject Classification: Primary: 35J20, 35J60, 47G20, 49J40.

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