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

  • Shapour Heidarkhani EMAIL logo , Ghasem A. Afrouzi and Shahin Moradi
Published/Copyright: February 8, 2018

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,

{div((1+|u|p(x)1+|u|2p(x))|u|p(x)2u)+α(x)|u|p(x)2u=λf(x,u)inΩ,uν=0onΩ

where ΩRN(N2) is a bounded domain with boundary of class C1,ν is the outer unit normal to Ω,λ>0, αL(Ω),f:Ω×RR is an L1-Carathéodory function and pC0(Ω). 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.

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Received: 2017-5-25
Accepted: 2018-1-15
Published Online: 2018-2-8
Published in Print: 2018-4-25

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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