 Solution

Every regular radial solution has outward normal derivative $u^{\prime }(a)=a/3$, which is nonzero since $a>0$. Therefore 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 requirewhich 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.