In the paper A remark on least energy solutions in $\mathbb{R}^N$, page 2407, it said, if $u_0\in H^1(\mathbb{R}^2)$, set $\gamma(t)=t^{-1/4}u_0(x/t)$. Then $\gamma(t)$ is a continuous path in $L^2(\mathbb{R}^2)\cap L^4(\mathbb{R}^2)$(but not in $H^1(\mathbb{R}^2)$) joining $0$ and $u_0$. How to prove it is continuous in $L^2(\mathbb{R}^2)\cap L^4(\mathbb{R}^2)$, but not in $H^1(\mathbb{R}^2)$? Why the factor added in the definition of $\gamma$ is $t^{-1/4}$?
Thank you!