Nonexistence of minimizers for sufficiently large charge

Let $q>0$ be the charge parameter, and consider the minimization problem defined by equation (minmin1), whose minimizers are configurations attaining its minimum energy. Nonexistence conjecture. There exists a threshold $M>0$ such that for $q>M$ no minimizer occurs for the minimization problem.

The conjecture is motivated by the large-$q$ energy estimates developed in the paper and by the proof of Lu and Otto for the Thomas–Fermi–Dirac–von Weizsäcker energy. It asserts nonexistence of minimizers beyond a universal charge threshold; the supplied source does not indicate that this has been proved or disproved.