A minimizing problem of a polyharmonic operator with Critical Exponent

TL;DR

This paper investigates minimization problems involving polyharmonic operators with critical exponents, proving the existence of minimizers and comparing two related infimum values under certain conditions.

Contribution

It establishes the existence of minimizers for specific polyharmonic minimization problems and demonstrates the strict inequality between two infimums for a broad class of functions.

Findings

Existence of minimizers when 4a5; eq 0.

Proved that S_{ heta,r}(4a5) < S_{0,r}(4a5) for many 4a5.

Analyzed minimization problems involving polyharmonic operators with critical exponents.

Abstract

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$.