Sobolev Inequalities I have problems with the following exercise using Sobolev Embedding Theorems.
Let $B_1(0) \in \mathbb{R}^3$ the unit ball and consider the functional
$$ F(u) = \int_{\omega} | \nabla u(x)|^2 + a|u(x)|^p dx
$$  
in $H^1(\omega)$, with $a \in \mathbb{R}$ and $p \in [1, 6]$.


*

*Using Sobolev embedding theorems, show that $F(u)$ is weakly lower semicontinous in $H^1$ if $ a \geq 0 $ or $ a \leq 0$ and $p \in [1,6)$.

*If $p= 6$ and $a < 0$ show that $F$ is strongly continous.

*Use the sequence $u_n= c n ^{\alpha} \max \{0, 1- n^2 |x|\}$ with a suitable choice od $c$ and $\alpha $ to show that $F$ is not weakly lower semicontinous when $a < 0$ and $p=6$.


I know that I should use the compactness of Sobolev embedding, but I cannot see how.
Thank you very much for every kind of help.

*

*if $p\in[1,6)$, we have $\nabla u_n\to \nabla u$ weakly in $L^2$ implies $u_n\to u$ strongly in $L^p$ (this is sobolev compact embedding). Hence, you have l.s.c. of $F$.

*if $u_n\to u$ strongly in $H^1$, it implies that $u_n\to u$ strongly in $L^6$. (Note that if $u_n\to u$ weakly in $H^1$, you only have $u_n\to u$ strongly in $L^p$ for $p<6$, like in $1.$)

*Check out Exercise 11.15 in this book.
