You will need to solve Poisson's equation (formula here) on account of each case r<a (easy), a<r<b, b<r<c, b<r (easy).
The potentials of three concentric spherical conductors of radii a,b,c such that a<b<c are held at the values
Solve the potential problem by assuming for a<r<b and b<r<c.
and continuity at r=b.
Calculate the total charge on the conductor of radius b .
and capacitance of the system.
You will need to solve Poisson's equation (formula here) on account of each case r<a (easy), a<r<b, b<r<c, b<r (easy).