# Positive splitting of Sobolev convergence

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 $$g_k = \frac{g}{f} f_k, \hspace{1cm} h_k = \frac{h}{f} f_k.$$