My question regards a minor detail in the proof of a lemma in a research paper I am reading.
Let $\Omega$ be an smooth, bounded open set of $\mathbb{R}^3$ and suppose $u_n \rightharpoonup u$ in $H_0^1 (\Omega)$. The author claims that since $H_0^1 (\Omega) \hookrightarrow L^q(\Omega)$ with compact embedding it follows that $u_n^2 \to u^2$ in $L^{6/5}(\Omega)$.
Why is this true?
By Sobolev embeddings, $u_n\rightharpoonup u$ in $L^p$ for all $p\le 6$ and $u_n\to u$ in $L^p$ for all $p<6$.
Write $$ u_n^2 - u^2 = (u_n-u)(u_n+u). $$ The first factor tends to zero in $L^p$, $p<6$, the second factor is bounded in $L^6$, so by Hoelder inequality, their product tends to zero in $L^q$ for all $q<3$. Since $6/5<3$ the claim follows.