Let $u\in H^1(\mathbb{R^n})$ be a weak solution of $-\Delta u+u=f$ in $\mathbb{R}^n$, where $f\in L^2(\mathbb{R}^n)$. I am trying to solve one problem and the first step should be to apply the Fourier transform to both members of the equation and I am a bit confused.
I know that I can apply Fourier transform to both members because $u, f$ are in $L^2$ and so are tempered distributions, and $\Delta u$ makes sense as a tempered distribution. Also, since $u$ is a weak solution I know that $$\int\nabla u\nabla v+\int uv =\int fv \enspace\enspace\enspace\forall v\in H^1_0.$$
Now my question is on how to apply Fourier transform to both members: do I write, for example, the FT of $f$ seeing $f$ as a distribution or a function of $L^2$? And is this what we get as the FT of $\Delta$:
$$\langle F(-\Delta u),\varphi \rangle = \langle-\Delta u, Fv \rangle = \sum_i \langle u_{x_i}, (Fv)_{x_i} \rangle = \sum_i \langle u_{x_i}, F(-2\pi i xv) \rangle =\int \nabla u F(-2\pi ixv)?$$
Thank you very much!