Assuming the electron to be a classical point particle, if one calculates the self-energy one finds $$U=\frac{e^2}{8\pi\epsilon_0r}$$ which diverges as $r\rightarrow 0$.
This formula is the result of calculation of electrostatic potential energy of a system of charged particles distributed at different points of space. The usual distribution employed for this calculation is a uniform surface density on a sphere of radius $r$, or uniform volume density in a ball of radius $r$. It makes no sense to do this calculation for a single charged particle existing at a point. So your claim above is incorrect; for classical point particle, we have no way to calculate its self-energy and arrive at this formula. The calculation has to be done for some system of particles, whose size is described by $r$.
Therefore, the measured mass of the electron should be $$m_{0e}+U/c^2=m_e.$$
Yes, this is the so-called electromagnetic mass effect; EM energy associated with internal EM interactions in a charged system (electrostatic potential energy) manifests as modification of its effective inertial mass. This can be positive or negative, in case the system parts have charge of same sign, it is positive.
Why is it a problem that in classical electrodynamics, the self-energy of a point electron diverge? The divergence in $U$ may be absorbed in $m_{0e}$ which is unobservable as often done in renormalization.
"Self-energy of a point electron diverges" is a result of unwarranted application of either 1) the Poynting energy formula to calculate EM interaction energy inside a point particle, or 2) trying to calculate the limit $\lim_{r\to 0} U$ and assuming this limit (which is infinity) is a valid result for the point particle. Both of these methods give infinite energy. But this merely says that either
- the Poynting energy formula value for field of point charge is infinite, or
- compressing spatially distributed charge to a point takes infinite energy.
But this does not imply all point charges have to be associated with infinite real energy (in the sense of stored previously done work, or extractable energy available for doing work later). Maybe point charge energy is not given by the Poynting formula, and maybe point charge is not a result of compression of spatially distributed charge into a point.
Is "Self-energy of a point electron diverges" a problem? It depends on other assumptions. If you regard Poynting energy formula as invalid for point particles, then this just the mathematical property of the Poynting energy of point particle, devoid of any physical relevance, so this is not a problem; electron energy is simply not given by the Poynting formula. This is closely related to the view that point particles do not act on themselves = no self-interaction in point particles. This viewpoint has been analyzed and developed into a consistent theory of charged point particles by many people in the past, such as Tetrode, Fokker, Frenkel, and the Feynman-Wheeler collaboration.
If you believe this infinite Poynting energy is real energy that had to be used in the past to assemble the point particle, or energy that can be released somehow, then the fact this energy is infinite is problematic, because it predicts potential infinitely strong bomb in every point particle. Also, there is no consistent theory of point particles with infinite EM self-energy; such theory has to involve non-EM energy, hence non-EM forces and also has to explain radiation energy as loss of self-energy through self-interaction. Such self-interaction of point particle cannot be consistently described in the framework of Maxwell equations and Lorentz force (so that local energy conservation and equations of motions are satisfied). One would have to modify the theory so that things would work much differently and EM self-interaction was somehow finite and consistent with radiation energy, while EM self-energy is infinite. This was tried many times but compelling solution was never found.
This is presented as a problem in textbooks because often one equates $m_ec^2=U$, and forget about $m_{e0}$. But I don't see a reason why $m_{e0}$ should be neglected. See page 28 here.
You are correct, the fact that non-EM forces have to be present to hold the charge together (and cancel the positive infinity of energy by negative infinite of energy) is sometimes omitted.
EM textbooks aren't usually very good on this topic. Probably because this problem with infinite EM energy and self-interaction of point charged particles has been sort-of solved long ago, but the solution (no self-interaction and Poynting formulae are not applicable to point particles) is so different from what people expected (consistent theory of self-interaction of point particles). Also no big revelation came from it. So the solution is not well known and accepted part of physics. This subject is still considered a murky mine-field area.
