The following is a claim in some lecture notes I am reading:
Consider a bounded sequence $(u_n) \subset H^1(\mathbb R^N)$ with $\|u_n\|_{L^2} \to \lambda > 0$. For $n \in \mathbb N$, the concentration functional is defined as $$ F_n(t) = \max_{y \in \mathbb R^N} \int_{B_t(y)} |u_n|^2 \ dx, \quad t \in [0, \infty). $$ It is clear that $F_n$ is a sequence of uniformly bounded and non-decreasing, non-negative functions. This sequence converges pointwisely.
The author gives a hint: The sequence $F_n$ is equicontinuous. For this, the Gagliardo-Nirenberg inequality may be useful.
My thought: With equicontinuity, we can apply the Ascoli-Arzelá theorem on $[0, 1)$ and obtain a subsequence that converges pointwisely in this interval. With this subsequence, we repeat the procedure on $[0, 2)$, and so on.
My questions: does this argument work? How to prove equicontinuity?
Gagliardo-Nirenberg inequality: Let $2 \leq q \leq \infty$ and $s > 0$ be such that $$ \frac 1 q = \frac 1 2 - \frac{\theta s}{d} $$ for some $\theta \in [0, 1)$. Then, for any $u \in H^s(\mathbb R^N)$ we have $$ \|u\|_{L^q(\mathbb R^N)} \leq C(N, q, s) \|u\|_{L^2(\mathbb R^N)}^\theta \|u\|_{\dot H^s(\mathbb R^N)}^{1 - \theta}, $$ where $\dot H^s$ is defined by $$ \|f\|_{\dot H^s}^2 = \int_{\mathbb R^N} |\xi|^{2s} |\widehat f(\xi)|^2 \ d \xi. $$