  In the Einstein-Maxwell-charged scalar system with a negative cosmological
constant in arbitrary dimensions higher than three, there exists a horizonless
charged soliton solution, which we construct explicitly for an arbitrary mass
of the scalar in perturbative series in small charge. We find that the
stability of the soliton is determined by the validity of the AdS weak gravity
conjecture. The existence of a stable soliton might endanger the convexity of
the (free) energy as a function of the charge because the phase transition
between the soliton and the extremal Reissner-Nordstrom black hole would be
discontinuous. We, however, argue that the existence of the hairy black hole
solution circumvents the violation of convexity. The thermodynamic properties
of the hairy black hole show that the phase transition becomes continuous
irrespective of whether the AdS weak gravity conjecture holds. When it holds,
the phase transition occurs between the soliton and the hairy black hole, and
when it is violated, the phase transition occurs between the extremal
Reissner-Nordstrom black hole and the hairy black hole.
