论文标题
具有关键指数的多谐操作员的最小化问题
A minimizing problem of a polyharmonic operator with Critical Exponent
论文作者
论文摘要
在这项工作中，我们研究了$ r \ in \ mathbb {n}^{*} $，\ begin {qore*} \ begin {array} {array} {ccc} s_ {0，r}（φ）= \ displaystyle \ inf_ { h_ {0}^{r}（ω）\，| u+φ\ | _ {l^{2^{*r}}} = 1} = 1} \ | U \ | _ {r}^{2}^{2}＆\ textrm {and}＆s_ {and}＆s_ {and}＆s_ {unfor_ {unfor_ {fistion fistion h_θ^{r}（ω）\，\ | U+φ\ | _ {l^{2^{*r}}} = 1} = 1} \ | u \ | _ {r}^{2}^{2} {2}，\ end enay} > 2r $，是一个平滑的界面域，$ 2^{*r} = \ frac {2n} {n-2 r} $，$φ\ in l^{2^{*r}}（r}}}（ω）\ cap c（ω）$和norm $ \ |。 \ | _ {r} = \ displayStyle {\int_Ω|（-Δ）^α。 \ | _ {r} = \ displaystyle {\int_Ω| \ nabla（-Δ）^α。 |^{2} dx} $其中$α= \ frac {r-1} {2} $如果$ r $是奇数。首先，我们证明，当$ s_ {0，r}（φ）$和$ s_ {θ，r}（φ）$中，$ s_ {0，r}（φ）$中的nimimum在$ s_ {0，$ equiv 0时证明。其次，我们证明了$ s_ {θ，r}（φ）<s_ {0，r}（φ）$，用于大型$φ$。
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$.
