A minimizing problem of a polyharmonic operator with Critical Exponent

Abstract

In this work, we study the two following minimization problems for $r \in \mathbb{N}^{*}$, \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 $Ω\subset \mathbb{R}^{N}, $ $N > 2r$, is a smooth bounded domain, $2^{*r}=\frac{2N}{N-2 r}$, $φ\in L^{2^{*r}} (Ω) \cap C(Ω)$ and the norm $\|. \|_{r}=\displaystyle{ \int_Ω |(-Δ)^α .|^{2}dx}$ where $ α=\frac{r}{2} $ if $r$ is even and $\|. \|_{r}=\displaystyle{ \int_Ω |\nabla(-Δ)^α . |^{2}dx }$ where $α= \frac{r-1}{2}$ if $r$ is odd. Firstly, we prove that, when $φ\not\equiv 0, $ the infimum in $S_{0,r}(φ)$ and $S_{θ,r}(φ)$ are attained. Secondly, we show that $ S_{θ,r}(φ)< S_{0,r}(φ) $ for a large class of $ φ$.