I have to solve this exercise:
"Fix $v \in \mathcal{C}^\infty_c(\mathbb{R})$. Discuss the strong and the weak convergence of the sequence $u_n(x) = \frac{v(nx-n^2)}{n}$ in the spaces $W^{k,p}(\mathbb{R})$."
My attempt is to calculate the $\| u^{(k)}_n \|_{W^{k,p}(\mathbb{R})}$ and see when it belongs to $L^{p}(\mathbb{R})$. In particular my result is when $p \le \frac{1}{k-1}$. Am I right?
Instead for the weak convergence: can I say, since the sequence converges strongly (for $p \le \frac{1}{k-1}$), it converges weakly?
You should give the condition in terms of $k$ and not of $p$. The sequence converges to the zero function strongly in $W^{k,p}$ when $k<1+1/p$ (that is, $k=0$ or $1$.)
However, the sequence converges weakly to $0$ for all $k$. To see this, let $\phi\in C_c^\infty(\Bbb R)$ be a test function. For large $n$, the supports of $\phi$ and $u_n$ are disjoint, so that $$ \int_{\Bbb R}u_n(x)\,\phi(x)\,dx=0 $$ for large enough $n$.