In this paper we consider a two dimensional cholesteric liquid crystal with an applied external field, whose field configurations \(u:\mathbb {R}^2\rightarrow S^2\) are constant at infinity, and can be classified by their topological degree. We look for non trivial, topologically stable configurations (chiral skyrmions) by minimizing the Oseen-Frank energy functional over the functions u with \(\deg (u)=-1\) . The energy functional depends on the splay ( \(K_1\) ), twist ( \(K_2\) ) and bend ( \(K_3\) ) elastic constants, and, assuming that \(K_1=K_2\) , we show the existence of skyrmions provided the cholesteric pitch \(q_0\) is small enough; moreover we study the compactness of such skyrmions as \(q_0\rightarrow 0\) , and their limit. Our results generalize some previous results in the literature obtained under the assumption that \(K_1=K_2=K_3\) (the well known “one-constant approximation” assumption). Moreover, in order to overcome the lack of compactness of the energy functional, we use (instead of the concentration-compactness principle) the fact that the field configurations with energy below a suitable threshold do not cover twice \(S^2\) minus a neighborhood of the North Pole, together with a truncation argument at infinity.
