Weak solutions for elliptic problems in weighted anisotropic Sobolev space

Abstract

Here, using the Mountain Pass theorem, the existence of weak solutions for the problem
\[
- \sum_{i=1}^{N} \frac{\partial}{\partial x_{i}}\left(a(x) | \frac{\partial u}{\partial x_{i}} |^{p_{i}(x)-2} \frac{\partial u}{\partial x_{i}} \right) = \lambda \gamma(x) |u|^{q(x)-2} u - \lambda \delta(x) |u|^{r(x)-2} u
\]
under the Dirichlet boundary conditions is studied.