Document Type : Research Paper

Authors

• Ahmed Ahmed ¹
• Mohamed Saad Bouh Elemine Vall ²

¹ Mathematics and Computer Sciences Department, University of Nouakchott, Faculty of Science and Technology, Nouakchott, Mauritania

² Department of Mathematics, Professional University Institute, Research unity: Modelling and Scientific Calculus, Nouakchott, Mauritania

Abstract

We consider in this paper a Neumann $\vec{p}(x)-$elliptic problems of the type
$$\left\{\begin{array}{ll}
- \Delta_{\vec{p}(x)} u+ \lambda(x)|u|^{p_{0}(x)-2}u = \alpha f(x,u)+ \beta g(x,u) \quad &\mbox{in} \quad \Omega, \\
\displaystyle\sum_{i=1}^{N}\Big| \frac{\partial u}{\partial x_{i}}\Big|^{p_{i}(x)-2}\frac{\partial u}{\partial x_{i}}\gamma_{i} =0 \quad &\mbox{on} \quad \partial\Omega.\end{array}\right.$$
We prove the existence of three weak solutions in the framework of anisotropic Sobolev spaces with variable exponent $W^{1,\vec{p}(\cdot)}(\Omega)$ under some hypotheses. The approach is based on a recent three critical points theorem for differentiable functionals.

Keywords

• Neumann elliptic problem
• weak solutions
• Variational principle
• Anisotropic variable exponent Sobolev spaces