A minimizing problem of a polyharmonic operator with Critical Exponent

Abstract

In this work, we study the two following minimization problems for $r\in {\mathbb{N}}^{\ast }$, \begin{equation*} \begin{array}{ccc} S_{0,r}(φ)=\displaystyle\inf_{u\in H_{0}^{r}(Ω)\,|u+φ\|_{L^{2^{*r}}}=1}\|u\|_{r}^{2}& \textrm{and}& S_{θ,r}(φ)=\displaystyle\inf_{u\in H_θ^{r}(Ω)\, \|u+φ\|_{L^{2^{*r}}}=1}\|u\|_{r}^{2}, \end{array} \end{equation*} where $\Omega \subset {\mathbb{R}}^N,$ $N>2r$, is a smooth bounded domain, $2^{\ast r}=\frac{2N}{N-2r}$, $\phi \in L^{2^{\ast r}}(\Omega )\cap C(\Omega )$ and the norm $\Vert .{\Vert }_r={\int }_{\Omega }|(-\Delta {)}^{\alpha }.{|}^{2}dx$ where $\alpha =\frac{r}{2}$ if $r$ is even and $\Vert .{\Vert }_r={\int }_{\Omega }|\nabla (-\Delta {)}^{\alpha }.{|}^{2}dx$ where $\alpha =\frac{r-1}{2}$ if $r$ is odd. Firstly, we prove that, when $\phi \not\equiv 0,$ the infimum in $S_{0,r}(\phi )$ and $S_{\theta ,r}(\phi )$ are attained. Secondly, we show that $S_{\theta ,r}(\phi )<S_{0,r}(\phi )$ for a large class of $\phi$.