I have obtained a Palais-Smale sequence $x_n$ for a functional $I:H^1(\mathbb{T}^2)\rightarrow \mathbb{R}$ (where $\mathbb{T}^2$ is a two dimensional torus), i.e. $I(x_n)\rightarrow c$ and $I'(x_n)\rightarrow 0$. This sequence is weakly convergent to some point $x\in X$. I want to prove that is strongly convergent. In order to do that, I just need the convergence of the norm, $\Vert x_n\Vert_{H^1}\rightarrow \Vert x\Vert_{H^1}$, in view of In Hilbert space: $x_n → x$ if and only if $x_n \to x$ weakly and $\Vert x_n \Vert → \Vert x \Vert$. . I want to know if my reasoning is correct:
Since $x_n\rightharpoonup x$ in $H^1$, $x_n\rightharpoonup x$ in $L^2$ and by Rellich-Kondrachov theorem, $x_n\rightarrow x$ in $L^2$ (up to a subsequence that we assume is the same). Similarly, $\nabla x_n\rightarrow\nabla x$ in $L^2$. Therefore, $\Vert x_n\Vert_{L^2}\rightarrow \Vert x\Vert_{L^2}$ and $\Vert \nabla x_n\Vert_{L^2}\rightarrow \Vert \nabla x\Vert_{L^2}$, which implies that $\Vert x_n\Vert_{H^1}\rightarrow \Vert x\Vert_{H^1}$. Is my reasoning correct? Any comment is welcome! Thank you.