Existence of solution for a anisotropic equation with critical exponent

Abstract

Using variational methods we establish existence of nontrivial solutions for the following class of anisotropic critical problem $$ {(P_{\lambda})} \qquad \left \{ \begin{array}{l} - \displaystyle \sum_{i=1}^{N} \frac{\partial}{\partial x_{i}} \Big ( \Big | \frac{\partial u}{\partial x_{i}} \Big |^{p_{i}-2}\frac{\partial u}{\partial x_{i}} \Big )= \lambda f(u) + g(u) , \quad \mbox{in} \quad \Omega \\ u \geq 0, \quad \mbox{in} \quad \Omega \\ u=0, \quad \mbox{on} \quad \partial \Omega , \end{array} \right. $$ where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$, $\lambda$ is a positive parameter, $g(u)$ behaves like $|u|^{p^*-2}u$, $p^{*}$ is the critical exponent for this class of problem and $f$ is a continuous function verifying some adequate assumptions.