Let $U = \{ x \in \mathbb{R}^3 | |x| < \pi \}$. Show that a necessary condition for:
$-\Delta u - u = f$
to have a weak solution in $H^1_0(U)$ is that:
$\int_{U} f(x) \frac{\sin(|x|)}{|x|} dx = 0$
From the look of this, I have guessed that we need to turn the PDE into spherical coordinates and then apply the Fredholm's alternative to complete the proof. The fact that $U$ is a sphere in $\mathbb{R}^3$ and the term $\sin(\cdot)$ kinda hints at that. However, I am kinda stuck there. Any suggestions?