Nonexistence for mixed-type equations with critical exponent nonlinearity in a ball

In this work, we consider the following isotropic mixed-type equations: (0.1)y|y|α−1uxx+x|x|α−1uyy=f(x,y,u) in Br(0)⊂R2 with r>0. By proving some Pohozaev-type identities for (0.1) and dividing Br(0) naturally into six regions Ωi(i=1,2,3,4,5,6), we can show that the equation (0.2)yuxx+xuyy=sign(x+y)|u|2u with Dirichlet boundary conditions on each natural domain Ωi has no nontrivial regular solution in Br(0).