I have a non-trivial weak solution $u_0 \in H^1(\mathbb{R}^N)$ for the following problem:
$$\begin{align*} (P_1) \begin{cases} -\Delta u + u &= \lambda f(u) \ \text{in} \ \mathbb{R}^N,\\ &u \in H^1(\mathbb{R}^N), \end{cases} \end{align*}$$
$\lambda > 0$
and I would like to show that $w(x) := u_0 \left( \frac{x}{t} \right)$, with $t > 0$, is a non-trivial weak solution for the problem:
$$\begin{align*} (P_2) \begin{cases} -\Delta u + u &= f(u) \ \text{in} \ \mathbb{R}^N,\\ &u \in H^1(\mathbb{R}^N), \end{cases} \end{align*}$$
I would like to know if my computation below is correct.
Let $y := \frac{x}{t}$.
$$\begin{align*} \int_{\mathbb{R}^N} \nabla w(x) \nabla v(x) dx &= \int_{\mathbb{R}^N} \nabla u_0 \left( \frac{x}{t} \right) \nabla v(x) dx\\ &= \int_{\mathbb{R}^N} \nabla u_0(y) \nabla v(ty) t^N dy\\ &= \int_{\mathbb{R}^N} \nabla u_0(y) t \nabla v(ty) t^N dy\\ &= t^{N+1} \int_{\mathbb{R}^N} \nabla u_0(y) \nabla v(ty) dy\\ &= t^{N+1} \lambda \int_{\mathbb{R}^N} f(u_0(y)) v(ty) dy\\ &= t^{N+1} \lambda \int_{\mathbb{R}^N} f \left( u_0 \left( \frac{x}{t} \right) \right) v(x) \frac{1}{t^N} dx\\ &= t \lambda \int_{\mathbb{R}^N} f(w(x)) v(x) dx, \forall v \in H^1(\mathbb{R}^N). \end{align*}$$
If $t = \frac{1}{\lambda}$, then $w$ is a non-trivial weak solution for $(P_2)$.
Thanks in advance!