Mathematica Bohemica, online first, 20 pp.

Existence of infinitely many weak solutions for some quasilinear $\vec{p}(x)$-elliptic Neumann problems

Ahmed Ahmed, Taghi Ahmedatt, Hassane Hjiaj, Abdelfattah Touzani

Received July 14, 2015.   First published January 2, 2017.

Ahmed Ahmed, Taghi Ahmedatt, Abdelfattah Touzani, Department of Mathematics, Laboratory LAMA, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, BP 1796 Atlas, Fez, Morocco, e-mail:,,; Hassane Hjiaj, Department of Mathematics, Faculty of Sciences, Tetouan University Abdelmalek Essaadi, Quartier M'haneche II, Avenue Palestine, BP 2121, Tetouan 93000, Morocco, e-mail:

Abstract: We consider the following quasilinear Neumann boundary-value problem of the type $ - \displaystyle\sum_{i=1}^N\frac{\partial}{\partial x_i}a_i\Big(x,\frac{\partial u}{\partial x_i}\Big) + b(x)|u|^{p_0(x)-2}u = f(x,u)+ g(x,u) &\text{in} \Omega, \quad\dfrac{\partial u}{\partial\gamma} = 0 &\text{on} \partial\Omega. $ We prove the existence of infinitely many weak solutions for our equation in the anisotropic variable exponent Sobolev spaces and we give some examples.

Keywords: Neumann problem; quasilinear elliptic equation; weak solution; variational principle; anisotropic variable exponent Sobolev space

Classification (MSC 2010): 35J20, 35J62

DOI: 10.21136/MB.2017.0037-15

Full text available as PDF.

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