Existence of 2D Skyrmions

In this paper, we consider the 2D Skyrme model $$E(u)=\frac{1}{2} \int\limits_{R^2}|du|^2dx+\frac{\lambda}{4} \int\limits_{R^2}|du\wedge du|^2dx+\frac{\mu}{16} \int\limits_{R^2}|u-{\bf{n}}|^4dx,$$ where ?? and??? > 0 are positive coupling constants and n = (0, 0, 1) is the north pole of S ². We derive a lower bound of 2D Skyrme model. Using this estimate, we prove the existence of 2D Skyrmion for any positive coupling constants ??, ??.