= Solution
Every regular radial solution has outward normal derivative $u'(a)=a/3$, which is nonzero since $a>0$. Therefore \b[there is no solution] with the stated homogeneous <Neumann boundary condition>. This also follows without imposing spherical symmetry: the compatibility condition from part (i) would require
$$
\frac{4\pi a^3}{3}=\int_V1\,dV=\int_S0\,dS=0,
$$
which is impossible. Thus the full-ball <Poisson equation> has no classical solution for these boundary data, radial or otherwise. A singular $1/r$ term could formally cancel the derivative at $r=a$, but would introduce a source at the origin and violate the problem's equation there.
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