Can I find a sequence $(f_j)_{j\in\Bbb{N}}\in C^{\infty}(\Bbb{R^+})$ such that : $$ \lim_{j\to\infty}\int^\infty_0 \big(\partial^2_r f_j+\frac{1}{r}\partial_r f_j+r^2f_j\big)^2 rdr=0$$ and $$ \sqrt{\int_{\Bbb{R^+}}|f_j(r)|^2 rdr}=1 $$ .
Thank you for any suggestion whatsoever.