| Contents |
| In this talk, we consider the attainability of a maximizing problem \begin{equation*} D:=\sup_{\|u\|_{H^{1,N}_\gamma}=1}\left(\|u\|_N^N+\alpha\|u\|_p^p\right),
\end{equation*}
where $N\geq 2$, $N0$ and $\|u\|_{H^{1,N}_\gamma}=\left(\|u\|_N^\gamma+\|\nabla
u\|_N^\gamma\right)^{\frac{1}{\gamma}}$.
The existence of a maximizer for $D$ is closely related to the exponent $\gamma$.
In fact, we show that the value
\begin{equation*}
\alpha=\alpha_*:=\inf_{\|u\|_{H^{1,N}_\gamma}=1}\left(\frac{1-\|u\|_N^N}{\|u\|_p^p}\right)
\end{equation*}
is a threshold in terms of the attainability of $D$. |
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