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

For the 2-dimensional anisotropic Sobolev inequality of the form ∫ R 2 | u | 6 d x d y ⩽ α ( ∫ R 2 u x 2 d x d y ) 2 ∫ R 2 | D x − 1 u y | 2 d x d y , it is proved that the sharp (smallest) positive constant α is exactly as 3 ( ∫ R 2 ϕ x 2 d x d y ) − 2 , where ϕ is a minimal action solution of ( u x x + | u | 4 u ) x = D x − 1 u y y .