Anisotropic elliptic problems; Robin boundary conditions existence and multiplicity

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.

Keywords

• Anisotropic elliptic problems
• Robin boundary conditions
• existence and multiplicity