Existence of three weak solutions for an anisotropic quasi-linear elliptic problem

Document Type : Research Paper


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


We consider in this paper a Neumann $\vec{p}(x)-$elliptic problems of the type
- \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.


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Volume 14, Issue 10
October 2023
Pages 85-93
  • Receive Date: 22 April 2023
  • Revise Date: 30 April 2023
  • Accept Date: 10 May 2023