Hopf boundary maximum principle violation for semilinear elliptic equations

We consider elliptic equations with non-Lipschitz nonlinearity − Δ u = λ | u | β − 1 u − | u | α − 1 u in a smooth bounded domain Ω ⊂ R n , with Dirichlet boundary conditions; here 0 < α < β < 1 . We prove the existence of a weak nonnegative solution which does not satisfy the Hopf boundary maximum principle, provided that λ is large enough and n > max { 2 , 2 ( 1 + α ) ( 1 + β ) / ( 1 − α ) ( 1 − β ) } .