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