Vrftsjtryk

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 $(u_n)_{ninmathbb{N}}$ 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 $Nge3$ and so, by Sobolev embedding theorem, $(u_n)_{ninmathbb{N}}$ converges weakly in $L^{2^*}(Omega)$ to $bar u$, where $2^*=frac{2N}{N-2}$. Now, it is clamed that:
$$forall vin H^1_0(Omega), int_Omega |u_n(x)|^{2^*-2}u_n(x) v(x)operatorname{d}xto int_Omega |bar u(x)|^{2^*-2}bar u(x) v(x)operatorname{d}x, nto infty.$$
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 $|u_n-u|_{2^*}rightarrow0,nrightarrowinfty$, a thing that we don't have here.
Obviously I'm missing something... can anyone help me to figure out what I'm missing?

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.

Let's use $Vert . Vert_{alpha}$ to denote the usual norm on $L^{alpha}(Omega)$ for every $1leq alpha leq infty $. It is sufficient to show that:
$$
Vert (u_n|u_n|^{{2^*-2}} -bar{u}|bar{u}|^{{2^*-2}})vVert_1to 0 text{ as } n to infty
$$
Now, for any $phiin C^{infty}_0(Omega)$:
$$
Vert (u_n|u_n|^{{2^*-2}} -bar{u}|bar{u}|^{{2^*-2}})vVert_1\
leq
Vert (u_n|u_n|^{{2^*-2}} -bar{u}|bar{u}|^{{2^*-2}})phiVert_1
+
Vert (u_n|u_n|^{{2^*-2}} -bar{u}|bar{u}|^{{2^*-2}}) (v - phi)Vert_1\
$$

The function $tto t|t|^{{2^*-2}}$ is Lipschitz on every subset of the form $[-M,M]$ where its Lipschitz constant is $(2^*-1)M^{{2^*-2}}$. This gives us the following estimate:
$$|a|a|^{{2^*-2}} -b|b|^{{2^*-2}}|
leq (2^*-1)(|a|+|b|)^{{2^*-2}}|a-b| qquad forall a,b in mathbb{R}
$$
Therefore, thanks to this relation and Hölder inequality we are able to prove:
begin{align}
Vert (u_n|u_n|^{{2^*-2}} &-bar{u}|bar{u}|^{{2^*-2}})phiVert_1
leq Vert phiVert_{infty} Vert u_n|u_n|^{{2^*-2}} -bar{u}|bar{u}|^{{2^*-2}}Vert_1 \
&leq Vert phiVert_{infty} Vert (2^*-1)(|u_n| + |bar{u}|)^{2^*-2} (u_n-bar{u}) Vert_1 \
&leq (2^*-1) Vert phiVert_{infty} Vert (|u_n| + |bar{u}|)^{2^*-2} Vert_{frac{2^*-1}{2^*-2}} Vert u_n-bar{u} Vert_{2^*-1}\
&= (2^*-1) Vert phiVert_{infty} Vert|u_n| + |bar{u}|Vert^{2^*-2}_{2^*-1} Vert u_n-bar{u} Vert_{2^*-1} \
&leq (2^*-1) Vert phiVert_{infty} (Vert u_nVert_{2^*-1} + Vert bar{u}Vert_{2^*-1})^{2^*-2} Vert u_n-bar{u} Vert_{2^*-1} \
&leq (2^*-1) Vert phiVert_{infty} (2C_1)^{2^*-2} Vert u_n-bar{u} Vert_{2^*-1}
end{align}
where $C_1=max { sup_{n} { Vert u_n Vert_{2^*-1} },Vert bar{u} Vert_{2^*-1} }$ is finite since $H^1_0(Omega)$ embeds continuously in $L^{2^*-1}(Omega)$ and $u_n$ weakly converges to $u$ in $H^1_0(Omega)$.
We have to estimate the other term now:
begin{align}
Vert (u_n|u_n|^{{2^*-2}} &-bar{u}|bar{u}|^{{2^*-2}})(v - phi)Vert_1
leq Vert u_n|u_n|^{{2^*-2}} -bar{u}|bar{u}|^{{2^*-2}}Vert_{frac{2^*}{2^*-1}} Vert v - phiVert_{2^*}\
&leq (Vert u_n|u_n|^{{2^*-2}}Vert_{frac{2^*}{2^*-1}} +Vert bar{u}|bar{u}|^{{2^*-2}}Vert_{frac{2^*}{2^*-1}}) Vert v - phiVert_{2^*}\
&= (Vert u_nVert^{2^*-1}_{2^*} +Vert bar{u}Vert^{2^*-1}_{2^*}) Vert v - phiVert_{2^*}\
&leq (2C_2)^{2^*-1} Vert v - phiVert_{2^*}
end{align}
Where $C_2:=max{ sup_{n}{Vert u_nVert_{2^*}},Vert bar{u}Vert_{2^*} }$ that is finite thanks to Sobolev embedding of $H^1_0(Omega)$ in $L^{2^*}(Omega)$ combined, once again, with the weak convergence of $u_n$ in $H^1_0(Omega)$.
$$
0leqlimsup_{n to infty} Vert (u_n|u_n|^{{2^*-2}} -bar{u}|bar{u}|^{{2^*-2}})vVert_1\
leq
limsup_{n to infty} (2^*-1) Vert phiVert_{infty} (2C_1)^{2^*-2} Vert u_n-bar{u} Vert_{2^*-1}
+\
+ limsup_{n to infty} (2C_2)^{2^*-1} Vert v - phiVert_{2^*}\
=
(2C_2)^{2^*-1} Vert v - phiVert_{2^*}
$$
using that $u_n$ converges to $bar{u}$ stongly in $L^{2^*-1}(Omega)$ due to Rellich-Kondrachov compact embeddings.
Since $Vert v - phiVert_{2^*}$ can be chosen arbitrarily small as $C^{infty}_0(Omega)$ is dense in $L^{2^*}(Omega)$, the thesis follows.