Existence results for elliptic equation involving polyharmonic operator and a critical growth

In this work, we study the two following minimization problems for \(r \in \mathbb {N}^{*}\) $r\in \mathbb{N}_{}^{\ast }$ , \(\begin{aligned} \begin{array}{ccc} S_{0,r}(\varphi )=\displaystyle \inf _{u\in H_{0}^{r}(\Omega ),\,\Vert u+\varphi \Vert _{L^{2^{*r}}}=1}\Vert u\Vert _{r}^{2}&\text { and }\,\,&S_{\theta ,r}(\varphi )=\displaystyle \inf _{u\in H_{\theta }^{r}(\Omega ),\,\Vert u+\varphi \Vert _{L^{2^{*r}}}=1}\Vert u\Vert _{r}^{2}, \end{array} \end{aligned}\) $$\begin{array}{c}\hfill \begin{array}{ccc}{\displaystyle S_{0,r}(\phi )=inf_{u\in H_0^r(\mathrm{\Omega }),\Vert u+\phi \Vert _{L_{}^{2_{}^{\ast r}}}^{}=1}^{}{\Vert u\Vert }_r^2}& \text{and}& {\displaystyle S_{\theta ,r}(\phi )=inf_{u\in H_{\theta }^r(\mathrm{\Omega }),\Vert u+\phi \Vert _{L_{}^{2_{}^{\ast r}}}^{}=1}^{}{\Vert u\Vert }_r^2,}\end{array}\end{array}$$ where \(\Omega \subset \mathbb {R}^{N}, \) $\mathrm{\Omega }\subset {\mathbb{R}}^N,$ \(N > 2r\) $N>2r$ , is a smooth bounded domain, \(2^{*r}=\frac{2N}{N-2 r}\) $2_{}^{\ast r}=\frac{2N}{N-2r}$ , \(\varphi \in L^{2^{*r}} (\Omega ) \cap C(\Omega )\) $\phi \in L_{}^{2_{}^{\ast r}}(\mathrm{\Omega })\cap C(\mathrm{\Omega })$ and the norm \(\Vert . \Vert _{r}=\displaystyle { \int _{\Omega } |(-\Delta )^{\alpha }.|^{2}dx}\) ${\displaystyle {\Vert .\Vert }_r={\int }_{\mathrm{\Omega }}{|{(-\mathrm{\Delta })}^{\alpha }.|}^{2}dx}$ where \( \alpha =\frac{r}{2} \) $\alpha =\frac{r}{2}$ if r is even and \(\Vert . \Vert _{r}=\displaystyle { \int _{\Omega } |\nabla (-\Delta )^{\alpha }. |^{2}dx }\) ${\displaystyle {\Vert .\Vert }_r={\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{(-\mathrm{\Delta })}^{\alpha }.|}^{2}dx}$ where \(\alpha = \frac{r-1}{2}\) $\alpha =\frac{r-1}{2}$ if r is odd. Firstly, we prove that, when \(\varphi \not \equiv 0, \) $\phi \not\equiv 0,$ the infimum in \(S_{0,r}(\varphi )\) $S_{0,r}(\phi )$ and \(S_{\theta ,r}(\varphi )\) $S_{\theta ,r}(\phi )$ are achieved. Secondly, we show that \( S_{\theta ,r}(\varphi )< S_{0,r}(\varphi ) \) $S_{\theta ,r}(\phi )<S_{0,r}(\phi )$ for a large class of \(\varphi \) $\phi$ .