In the proof of the Brezis-Nirenberg theorem on the book "Nonlinear Analysis and Semilinear Elliptic Problems" by Ambrosetti and Malchiodi, it is used lemma 11.11 at pag. 183, in whose proof we have a sequence $(u_n)_{n\in\mathbb{N}}$ that converges to $\bar u$ in the following senses:
- weakly in $H^1_0(\Omega)$;
- almost everywhere;
where $\Omega$ is a non-empty bounded open subset of $\mathbb{R}^N$ with $N\ge3$ and so, by Sobolev embedding theorem, $(u_n)_{n\in\mathbb{N}}$ converges weakly in $L^{2^*}(\Omega)$ to $\bar u$, where $2^*=\frac{2N}{N-2}$. Now, it is clamed that:
$$\forall v\in H^1_0(\Omega), \int_\Omega |u_n(x)|^{2^*-2}u_n(x) v(x)\operatorname{d}x\to \int_\Omega |\bar u(x)|^{2^*-2}\bar u(x) v(x)\operatorname{d}x, n\to \infty.$$
I tried to figure out why it is so, using Hölder inequality or trying to find a dominating function like in the proof of Riesz-Fischer theorem, but failed in both cases… it seems to me that what I actually need to prove such a claim using that instruments it is that $\|u_n-u\|_{2^*}\rightarrow0,n\rightarrow\infty$, a thing that we don't have here.
Obviously I'm missing something… can anyone help me to figure out what I'm missing?
Best Answer
Edit The proof originally incorrectly applied a theorem concerning linear operators to a non-linear one, which has been fixed in the edit.
You need to exploit the weak convergence of $u_n.$ Observe that since $|(|u_n|^{2^*-2}u_n)| = |u_n|^{2^*-1},$ for each $n$ we get, $$|u_n|^{2^*-2}u_n \in L^{\frac{2^*}{2^*-1}}(\Omega) = L^{\frac{2N}{N+2}}.$$ Moreover the sequence $|u_n|^{2^*-2}u_n$ is uniformly bounded in $L^p$ where $p = \frac{2N}{N+2} > 1,$ so by reflexivity we get a weakly convergent subsequence $$|u_{n_k}|^{2^*-2}u_{n_k} \rightharpoonup v \in L^p(\Omega).$$ Since it also convergences a.e., we get $v = |\overline u|^{2^*-2}\overline u.$ As the above holds for every subsequence, we conclude that the entire sequence $|u_n|^{2^*-2}u_n$ converges weakly to $|\overline u|^{2^*-2}\overline u$ in $L^p(\Omega).$
As the conjugate dual of $p=\frac{2N}{N+2}$ is $2^*,$ for all $v \in L^{2^*}(\Omega)$ we have, $$ \int_{\Omega} |u_n|^{2^*-2}u_n v \,\mathrm{d} x \rightarrow \int_{\Omega} |\overline u|^{2^*-2}\overline u v \,\mathrm{d}x. $$ As $H^1_0(\Omega) \hookrightarrow L^{2^*}(\Omega)$ by Sobolev embedding, the result follows.