Let $s\in C(\mathbb R^d)$ and $s_n\in C^\infty(\mathbb R^d)$ be positive functions, both in $L^2(\mathbb R^d)$ for every $n\in\mathbb N$. Suppose that $$ s_n \,\to\,s \ \textrm{ in }L^2(\mathbb R^d)\ \textrm{ as }n\to\infty\;,$$ and there exists $c\in(0,\infty)$ such that $$ \int |\nabla s_n\,|^2 \,\leq\, c \quad \forall\,n\in\mathbb N \;.$$
Can we conclude that $s\in W^{1,2}(\mathbb R^d)$ ?
And, if yes, can we say that $\nabla s_n \to \nabla s\ $ in $L^2(\mathbb R^d)\ $ as $n\to\infty $ ?