Weak convergence in $H^1_0(\Omega)$ and convergence of integrals: an analysis of common statements

Question

Let $\Omega \subset \mathbb{R}^N$, with $N>2$, be a bounded domain and let $2^*$ be the critical Sobolev exponent given by $$2^* := \dfrac{2N}{N-2}.$$ If $(u_n)$ is a bounded sequence in $H^1_0(\Omega)$, we can guarantee the existence of a subsequence $(u_{n_j})$ such that $u_{n_j} \rightharpoonup u$ in $H^1_0(\Omega)$. Under these conditions, we cannot affirm that $$\int_{\Omega} |u_{n_j}|^{2^* - 2}u_{n_j}v \to \int_{\Omega} |u|^{2^* - 2}uv$$ for every $v \in H^1_0(\Omega)$?

Answer 1

Let us assume that $u_n \rightharpoonup u$ in $H^1_0(\Omega)$, $u_n \to u$ in $L^p$ with $p = \frac{N+2}{N-2}=2^*-1$ (which is smaller than the critical exponent).

If $v\in L^\infty(\Omega)$ then $$ \int_\Omega |u_n|^{2^*-2}u_n \cdot v \to \int_\Omega |u|^{2^*-2}u \cdot v $$ due to the strong convergence of $u_n$ in $L^p$.

Now take $v\in H^1_0(\Omega)$, $w\in C^\infty_c(\Omega)$. Then $$ \int_\Omega( |u_n|^{2^*-2}u_n- |u|^{2^*-2}u) \cdot v =\int_\Omega( |u_n|^{2^*-2}u_n- |u|^{2^*-2}u) \cdot (v-w) +\int_\Omega( |u_n|^{2^*-2}u_n- |u|^{2^*-2}u) \cdot w. $$ Due to density, we can choose $w$ such that the first integral is smaller than some $\epsilon>0$: $$ \left| \int_\Omega( |u_n|^{2^*-2}u_n- |u|^{2^*-2}u) \cdot (v-w) \right| \le 2^p ( \|u_n\|_{L^p}^p + \|u\|_{L^p}^p) \|v-w\|_{L^p} \le c ( \|u_n\|_{L^p}^p + \|u\|_{L^p}^p) \|v-w\|_{H^1} \le \epsilon. $$ With $w$ fixed, we can pass to the limit in the second integral to obtain $$ \limsup_{n\to\infty} \left| \int_\Omega( |u_n|^{2^*-2}u_n- |u|^{2^*-2}u) \cdot v \right|\le \epsilon. $$