Electron. J. Differential Equations, Vol. 2017 (2017), No. 188, pp. 1-17.

Existence and multiplicity of solutions for anisotropic elliptic problems with variable exponent and nonlinear Robin boundary conditions

Brahim Ellahyani, Abderrahmane El Hachimi

Abstract:
This article presents sufficient conditions for the existence of solutions of the anisotropic quasilinear elliptic equation with variable exponent and nonlinear Robin boundary conditions,
$$\displaylines{ -\sum_{i=1}^{N}\frac{\partial }{\partial x_{i}} \Big(\big| \frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)-2} \frac{\partial u}{\partial x_{i}}\Big) +\sum_{i=1}^{N}| u|^{p_{i}(x)-2}u+\lambda| u|^{m(x)-2}u =\gamma g(x,u)\quad \text{in } \Omega,\cr \sum_{i=1}^{N}\big| \frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)-2} \frac{\partial u}{\partial x_{i}}\upsilon_{i} =\mu| u|^{q(x)-2}u \quad\text{on } \partial\Omega. }$$
Under appropriate assumptions on the data, we prove some existence and multiplicity results. The methods are based on Mountain Pass and Fountain theorems.

Submitted May 24, 2016. Published July 24, 2017.
Math Subject Classifications: 35J20, 35J25, 35J62.
Key Words: Anisotropic elliptic problems; Robin boundary conditions existence and multiplicity.

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Brahim Ellahyani
Department of Mathematics
Faculty of Sciences
Mohammed V University
P.B. 10014, Rabat, Morocco
email: ellahyani.brahim.1991@gmail.com
Abderrahmane El Hachimi
Department of Mathematics
Faculty of Sciences
Mohammed V University
P.B. 10014, Rabat, Morocco
email: aelhachi@yahoo.fr

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