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Let $f,g,h \in H^1(\mathbb{R}^n)$ be non-negative Sobolev functions wuch that $f^2 = g^2 + h^2$. Let also $\{f_k\} \subseteq H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f_k \to f$ w.r.t. the strong $H^1$ topology. Is it true that there exist $g_k, h_k$ such that $f_k^2 = g_k^2 + h_k^2$, and $g_k \to g$, $h_k \to h$?

Note: if we state the analogous problem for $L^2(\mathbb{R}^n)$, then the answer is yes, letting $$ g_k = \frac{g}{f} f_k, \hspace{1cm} h_k = \frac{h}{f} f_k. $$

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