I have encountered the following issue on a homework question. There seems to be a gap in my knowledge and I cannot answer it. For the set up, let $\Omega \subset \mathbb{R}^{n}$ be a bounded domain with smooth boundary. Let $X$ be the Banach space defined by $H^1_0(\Omega) \cap L^{q+1}(\Omega)$ where $q+1 > 2^{\star}$ and $2^{\star} = 2n/(n-2)$ with norm $\| \cdot \|_{X} = \| \cdot \|_{H^1_0} + \| \cdot \|_{L^{q+1}}$. Let $2^{\star} \leq p+1<q+1$, can we say anything about the compactness of the embedding $X \hookrightarrow L^{p+1}(\Omega)$? In particular, is there any chance that $X \subset \subset L^{p}(\Omega)$ is compact?
This seems outside the jurisdiction of the usual embedding $H^1_0(\Omega) \subset \subset L^{r}(\Omega)$ for all $1 \leq r < 2^{\star}$. Basically what I'm trying to achieve is that $\| v_j \|_{L^{p+1}} \rightarrow \| v \|_{L^{p+1}}$ whenever $v_{j}$ converges to $v$ weakly in $X$.
Many thanks in advance!