Your region is in cartesian coordinates, while your function V is spherical coordinates. That is a mismatch. I tried to solve your equation in cartesian coordinates, but I have run into shortcomings of the current computational geometry:

c1 = Ball[{0, 0, 0}, 1];
c2 = Cone[{{0, 0, 0}, {0, 0, 1/2}}, 1/4];
\[CapitalOmega] = DiscretizeRegion@RegionDifference[c1, c2];
bdry = DiscretizeRegion@RegionBoundary[\[CapitalOmega]];
DiscretizeRegion@RegionIntersection[bdry, Ball[{0, 0, 0}, 9/10]]

fails with the message "DiscretizeRegion was unable to discretize the region RegionIntersection[<<2>>]"