Multiplicity of Positive Solutions for an Anisotropic Problem via Sub-Supersolution Method and Mountain Pass Theorem

We use the sub-supersolution method and the Mountain Pass Theorem in order to show existence and multiplicity of solution for an anisotropic problem given by $$\begin{cases} \ -\Big[\displaystyle\sum^{N}_{i=1}\frac{\partial}{\partial x_{i}} \Big( \Big\vert \frac{\partial u}{\partial x_{i}}\Big\vert^{pi-2} \frac{\partial u}{\partial x_{i}}\Big )\ \Big]=a(x)u+ h(x,u) \text{ in } \Omega\text{,}\\[1mm] \ u>0\text{ in }\Omega, \quad u=0\text{ on } \partial\Omega\text{.} \end{cases}$$ We also prove the uniqueness of the solution for the linear anisotropic problem, a Comparison Principle for the anisotropic operator and a regularity result.