Existence results for elliptic equation involving polyharmonic operator and a critical growth
Résumé
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 achieved.
Secondly, we show that $ S_{\theta,r}(\varphi)< S_{0,r}(\varphi) $ for a large class of $\varphi$.
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