BPS skyrmions and near-BPS skyrmions in dimension two

We consider a variant of the baby Skyrme model where the σ-model term is multiplied by a small parameter (Formula presented.), and with a quite general potential term. For (Formula presented.) this model becomes the BPS (Bogomol'nyi-Prasad-Sommerfield) model in two dimensions, with area-preserving diffeomorphism invariance and zero binding energies, and its soliton solutions are called BPS skyrmions. The solutions of the model where μ is nonzero but small are called near-BPS skyrmions, and in this paper we study, by variational methods, the relation between the existence of BPS skyrmions and near-BPS skyrmions. In particular, we show that the existence of a BPS skyrmion implies the existence of a baby skyrmion provided the parameter μ is small enough; moreover such near-BPS skyrmions converges to a BPS skyrmion modulo translations.