Hi I'm stuck with the proof of a concentration-compactness lemma. We have the following equation in $\mathbb{R}^N, N \ge 3$: $$ -\Delta u +u=|u|^{p-2}u, $$ where $2 < p < 2^{*}$.

The functional associated to that equation is given by $$ J(u) = \frac{1}{2}\|u\|^2_{H^1}-\frac{1}{p}\|u\|^p_{L^p}. $$ Because the Sobolev imbedding $H^1 (\mathbb{R}^N) \subset L^q (\mathbb{R}^N), 2 < q < 2^{*}$ is not compact, $J$ does not satisfy the Palais-Smale condition. So one considers the family $J_k$ of functionals defined by $$ J_k(u) = \frac{1}{2}\|u\|^2_{H^1 (B_k)}-\frac{1}{p}\|u\|^p_{L^p(B_k)}, $$ where $(B_k)_k$ is an open cover of $R^N$; $k$ positive integer.

Each $J_k$ now satisfies the PS condition.

The "lemma" says the following: Let $u_k \in H^1_0(B_k)$ be uniformly bounded in $H^1(\mathbb{R}^n)$, i.e. $\|u_k\| \le \Lambda$, with $\Lambda>0$ independent of $k$, and such that $J'(u_k) \to 0$ as $k \to \infty$. Then, along a subsequence, one of the following holds true:

- either $u_k \to 0$ in $H^1(\mathbb{R}^N)$,
- or, there exist $r,\delta>0$, and a sequence $a_k$ in $R^N$ such that $$ \liminf_k \int_{B_{r} (a_k)}u^2_k \ge \delta. $$ I know how to prove that if 2. does not hold then 1. holds. I need a hint on how to prove that if 1. does not hold then 2. holds.

Thanks