Uniqueness of Nash equilibrium in continuous two-player weighted potential games

The literature results about existence of Nash equilibria in continuous potential games [15] exploit the property that any maximum point of the potential function is a Nash equilibrium of the game (the vice versa being not true) and those about uniqueness use strict concavity of the potential function. The following question arises: can we find sufficient conditions on the potential function which guarantee one and only one Nash equilibrium when such a function is not strictly concave and the existence of a maximum is not ensured? The paper positively answers this question for two-player weighted potential games when the strategy sets are (not necessarily finite dimensional) real Hilbert spaces. Illustrative examples in finite dimensional spaces are provided, together with an application in infinite dimensional ones where a weighted potential function with a bilinear common interaction term is involved.