# Relationship between two minimization problems

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?