Extremal functions for the anisotropic Sobolev inequalities
Résumé
The existence of multiple nonnegative solutions to the anisotropic critical problem \begin{equation*} - \sum_{i=1}^{N} \frac{\partial }{\partial x_i} \left( \left| \frac{\partial u}{\partial x_i} \right|^{p_i-2} \frac{\partial u}{\partial x_i} \right) = |u|^{p^*-2} u \;\; \mbox{in} \;\; \mathbb{R}^N \end{equation*} is proved in suitable anisotropic Sobolev spaces. The solutions corres\-pond 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 $\calS$ is included in $L^\infty(\R^N)$ and is located outside of a ball of radius $\tau >0$ in $L^{p^*}(\R^N)$.
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