Let $1 \le p < n$ and $p^* = np/(n - p)$. Let $B \subset \mathbb{R}^n$ be a closed ball and let $\Omega \subset \mathbb{R}^n$ be an open set containing $B$. We denote by $W^{1, p}_{B}(\Omega)$ the set of nonnegative functions in $W^{1, p}(\Omega)$ such that $u = 1$ a.e. on $B$. Given $\lambda > 0$, we consider the two minimization problems \begin{gather} \inf \left\{\int_\Omega |u|^{p^*} \, dx : u \in W^{1, p}_B(\Omega), \, \|\nabla u\|_{L^p(\Omega; \mathbb{R}^n)} \le \lambda \right\}, \\ \inf \left\{a \ge 0 : u \in W^{1, p}_B(\Omega), \, u - a \in W^{1, p}_0(\Omega), \, \|\nabla u\|_{L^p(\Omega; \mathbb{R}^n)} \le \lambda\right\}. \end{gather}
Hence, the first problem seeks to minimize the $L^{p^*}$ norm and the second problem the boundary value among functions that are equal to $1$ in $B$. The gradient bound restricts the functions from decreasing too rapidly, so that the infima are not necessarily $0$.
Consider an increasing sequence $(\Omega_m)$ of bounded convex sets whose union is $\mathbb{R}^n$. Denote by $I_m$ and $a_m$ the infima of the two problems on $\Omega_m$ respectively. I would like to know the following: Can we prove $I_m \gtrsim a^{p^*}_m|\Omega|$ for large $m$? Or more generally, does the knowledge of $a_m$ deliver any lower bounds on $I_m$ asymptotically?
