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Nonexistence for mixed-type equations with critical exponent nonlinearity in a ball. (English) Zbl 1213.35317
Summary: We consider the following isotropic mixed-type equations:
\[ y|y|^{\alpha-1}u_{xx}+ x|x|^{\alpha-1}u_{yy}= f(x,y,u) \]
in \(B_r(0)\subset\mathbb R^2\) with \(r>0\). By proving some Pohozaev-type identities and dividing \(B_r(0)\) naturally into six regions \(\Omega_i\) \((i=1,2,3,4,5,6)\), we can show that the equation
\[ yu_{xx}+xu_{yy}= \text{sign}(x+y)|u|^2u \]
with Dirichlet boundary conditions on each natural domain \(\Omega_i\) has no nontrivial regular solution in \(B_r(0)\).

MSC:
35M12 Boundary value problems for PDEs of mixed type
35B44 Blow-up in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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