Sharp constant for a 2D anisotropic Sobolev inequality with critical nonlinearity

For the 2-dimensional anisotropic Sobolev inequality of the form∫R2|u|6dxdy⩽α(∫R2ux2dxdy)2∫R2|Dx−1uy|2dxdy, it is proved that the sharp (smallest) positive constant α   is exactly as 3(∫R2ϕx2dxdy)−2, where ϕ   is a minimal action solution of (uxx+|u|4u)x=Dx−1uyy.