Extremal functions for the anisotropic Sobolev inequalities

The existence of multiple nonnegative solutions to the anisotropic critical problem - \sum_{i=1}^{N} \frac{\partial}{\partial x_i} (| \frac{\partial u}{\partial x_i} |^{p_i-2} \frac{\partial u}{\partial x_i}) = |u|^{p^*-2} u {in} $\mathbb{R}$^N is proved in suitable anisotropic Sobolev spaces. The solutions correspond to extremal functions of a certain best Sobolev constant. The main tool in our study is an adaptation of the well-known concentration-compactness lemma of P.-L. Lions to anisotropic operators. Futhermore, we show that the set of nontrival solutions $\text{calS}$ is included in $L^{\mathrm{\infty }}({\mathbb{R}}^N)$ and is located outside of a ball of radius $\tau >0$ in $L^{p^{\ast }}({\mathbb{R}}^N)$.