Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty} \sup \{r : B(0, r) \subset \Omega_m \} = \infty. $$ Assume $u_m \in W^{1, p}(\Omega_m)$, $0 \le u \le 1$ and $$ \lim_{m \to \infty} \frac{1}{|\Omega_m|} \int_{\Omega_m} |u_m|^{p^*} \, dx = 0, \quad \|\nabla u_m\|_{L^p(\Omega_m, \mathbb{R}^n)} \le C, $$ where $C$ does not depend on $m$. Since $(u_m)$ is uniformly bounded in $L^\infty(\mathbb{R}^n)$, we can assume without loss of generality that there exists $u \in \dot{W}^{1, p}(\mathbb{R}^n) \cap L^\infty(\mathbb{R}^n)$ such that $u_m \overset{\ast}\rightharpoonup u $ in $L^\infty(\mathbb{R}^n)$. Is it true that $$\lim_{m \to \infty} \frac{1}{|\Omega_m|} \int_{\Omega_m} |u|^{p^*} \, dx = 0 $$ as well?

EDIT: We actually have more than weak$^*$ convergence. By the Rellich-Kondrachov theorem and the uniform boundedness of $(u_m)$ in $L^\infty$, it is true that $u_m \to u$ in $L^q(K)$ for all compact $K$ and all $1 \le q < \infty$.