Let $V\in L^p(\mathbb{R}^p)$ ($p\geq 1$) be compactly supported, non-negative and spherically symmetric.
Does there exist a unique weak solution for $$ \begin{cases} \left[-\Delta + V \right]f =0, \\ \lim_{\vert x \vert \rightarrow \infty} f(x)=0 \end{cases} $$ and what can be said about the regularity of $f$?
Any reference would be highly appreciated.
THIS IS A COMMENT
This is how I thought of proving that solutions are radially symmetric. There are a couple points where I am not entirely sure, though.
Anyway, the idea is to write $x=r\omega$, with $r=|x|$ and $\omega \in \mathbb S^{p-1}$, and decompose $f$ in spherical harmonics: $$f=\sum_\ell c_\ell(r) Y_\ell(\omega). $$ The task is to prove that $c_\ell=0$ unless $\ell=0$. The Laplacian is decomposed in a radial and in an angular part: $$-\Delta = -\Delta_r -\frac{1}{r^2}\Delta_{\omega}, $$ and since $V$ is radial, the equation $[-\Delta + V]f=0$ is equivalent to the infinite system of ODEs $$\tag{1} (-\Delta_r +V +\frac{\ell(\ell+d-2)}{r^2})c_\ell(r)=0,\quad \ell=0, 1, 2,\ldots$$ (The number $\ell(\ell+d-2)$ comes out from $-\Delta_\omega Y_\ell = \ell(\ell+d-2)Y_\ell$, but this is not important. What is important is that this number is nonnegative and that it vanishes if and only if $\ell=0$).
Since the solution must be smooth at $0$ (actually I am not so sure this is true since $V$ is not smooth itself), we have the two boundary conditions $$ c'_\ell(0)=0,\quad \lim_{r\to \infty} c_\ell(r)=0, $$ which are incompatible with (1) unless $c_\ell=0$ by some form of the maximum principle (and here I am also not entirely sure).