Earnshaw's theorem states that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges. This was first proven by British mathematician Samuel Earnshaw in 1842. It is usually referenced to magnetic fields, but was first applied to electrostatic fields. Earnshaw's theorem applies to classical inverse-square law forces (electric and gravitational).
It means that it is linked to the Gauss Law. And here a proof.
