It competes with the angular momentum contribution that is given by the expression l(l+1)/(2r2). So the effective potential would be (l2+l-2)/(2r2), that would be repulsive for l > 2 and attractive for 1 < l < 2. Since l is an integer, it can't be attractive and, for l=1 or 2 it will vanish. Similar considerations apply for an arbitrary coupling constant.

To solve for the eigenvalues and eigenfunctions, one, either writes the wavefunction as a series and demands that the series is a polynomial, which leads to the energy eigenvalues, as for the Coulomb potential, or one remarks that 1/r^2 is (up to a constant) the Green function on the sphere in four spatial dimensions, http://people.ds.cam.ac.uk/rc476/methods/sphintgf.v1.pdf and adapts the solution for the 1/r potential by Fock and Pauli to this case; cf. http://www.physics.drexel.edu/~bob/LieGroups/LG_14.pdf

Really good question. As you have asked, I can compare with coulomb case. We see that -1/r2 will approach X axis more rapidly than -1/r. When centripetal term is added, a minimum will be generated in the shape of the effective potential. This well will be narrower than Coulomb case. The figure is taken from https://physics.stackexchange.com/questions/314432/effective-potential-of-radial-equation-of-hydrogen

First, this potential is still a central force potential, which meets the radial equation just like that of The infinite spherical well ! You can find it in any textbook of Quantum Mechanics! The radial equation is

d2u(r)/dr2=(a/r2-k2)u,

where a=ll(l+1)-b, b=2mc/\hbar^2, k=\sqrt{2mE}/hbar, and c is the constant of the potential: V(r)=-c/r^2.

The solution depends on "a". As mentioned above by Stam Nicolis, if a=0, that is b=l(l+1), then the solution is Bessel function just like sin(r)/r. But for a general "a", especially beyond the integer values, the solutions are complicated! Surely, this can be solved anyway!

This potential is just like the centrifugal force potential which generates by the rotational motion the particle!

I hope this will be useful for you, good night!

The Schrödinger equation describes the volume within which the electron stabilizes in mostly stochastic axial resonance state, while the Coulomb force acting between the proton and the electron causes the energy of the electron to vary according to 1/r^2 as it axially resonates within this volume:

This paper should shed new light on this issue:

https://www.omicsonline.org/open-access/on-adiabatic-processes-at-the-elementary-particle-level-2090-0902-1000177.pdf