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 − 1 u x x + x | x | α − 1 u y y = f ( x , y , u ) in B r ( 0 ) ⊂ R 2 with r > 0 . By proving some Pohozaev-type identities for (0.1) and dividing B r ( 0 ) naturally into six regions Ω i ( i = 1 , 2 , 3 , 4 , 5 , 6 ) , we can show that the equation (0.2) y u x x + x u y y = sign ( x + y ) | u | 2 u with Dirichlet boundary conditions on each natural domain Ω i has no nontrivial regular solution in B r ( 0 ) .