Proof of a concentration compactness lemma

Question

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:

1. either $u_k \to 0$ in $H^1(\mathbb{R}^N)$,
2. 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

Answer 1

The concentration-compactness lemma of P.L. Lions is independent of the fact your sequence is Palais-Smale. It holds for general sequences of bounded positive measure, in your case $u_k^2\; dx$.

First normalize your sequence such that $\Vert u_k^2\Vert_2 =1$.

Hence the idea is to consider the concentration function

$$Q_k(r)=\sup_{x\in R^n} \left( \int_{B(x,r)} u_k^2 \;dx\right).$$

Note that $Q_k$ is non-decreasing and non-negative bounded function on $[0,+\infty[$, with $\displaystyle \lim_{r\rightarrow +\infty }Q_k(r)=1$.

Then there exists a subsequence of $Q_k$, still denote $Q_k$, and a non-decreasing and non-negative bounded function $Q$ on $[0,+\infty[$ such that

$$Q_k(r) \rightarrow Q(r),$$

for almost all $r$. We can assume that $Q$ is left-continuous and moreover since $Q_k$ is non-decreasing and bounded, we have $$Q(r)\leq \liminf_{k\rightarrow +\infty} Q_k(r) \hbox{ for all } r.$$

Let

$$\lambda =\lim_{r\rightarrow +\infty} Q(r).$$

If $\lambda=0$, this is the vanishing case, else let $r_0$ such that

$$Q(r_0)\geq \frac{\lambda}{2}.$$

For any $k$ there exists $x_k$ such that

$$Q_k(r_0)\leq \int_{B(x_k,r_0)} u_k^2 \;dx + \frac{1}{k}.$$

Hence for $k$ big enough, you have

$$\int_{B(x_k,r_0)} u_k^2 \;dx\geq \frac{\lambda}{3}.$$

In fact, you can refine this case in two

2.1) For all $\epsilon$ there exits $R$ such that $$\int_{B(x_k,R)} u_k^2 \;dx\geq 1-\epsilon.$$ This is the compactness-case.

2.2) the measure split on disjoint set arbitrary far, this is the dichotomy case.

You will find all the details in the two papers of Lions.