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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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References:
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