J. Nonl. Evol. Equ. Appl. 2019 (2), pp. 15-34, published on October 25, 2019:

Infinitely many solutions for an elliptic Neumann problem in Weighted variable exponent Sobolev spaces


A. Ahmed

University of Sidi Mohamed Ibn Abdel lah, Faculty of Sciences Dhar El Mahraz, Laboratory LAMA, Department of Mathematics; B.P. 1796 Atlas Fez, Morocco

Y. Akdim

University Sidi Mohamed Ben Abdellah, Faculty Polydisciplinary of Taza, Laboratory LSI, Department of Mathematics, Physics and Informatics; B.P 1223 Taza, Morocco

A. Touzani

University of Sidi Mohamed Ibn Abdel lah, Faculty of Sciences Dhar El Mahraz, Laboratory LAMA, Department of Mathematics; B.P. 1796 Atlas Fez, Morocco

Received on September 25, 2017 ; Final version: February 6, 2019
Accepted on February 2, 2019

Communicated by Khalil Ezzinbi

Abstract.  In this paper, we consider the Neumann elliptic problems of the type
Σi=1N ∂i ( wi(x) | ∂iu(x) |p(x) − 2 u(x) ∂iu(x) ) + w0(x) | u(x) | p(x) − 2 u(x) = f(x, u) + g(x, u) in Ω,
Σi=1N wi(x) | ∂iu(x) |p(x) − 2 u(x) ( ∂iu(x) ) γi(x) = 0 on ∂Ω.
We prove the existence of infinitely many weak solutions in the weighted variable exponent Sobolev spaces W1,p(·)(Ω, w), which generalizes the corresponding result of the reference [8].
Keywords:Neumann problem, variational principle, weighted variable exponent Lebesgue-Sobolev spaces.
2010 Mathematics Subject Classification:   35J20, 35J60, 47J10, 46E35.

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