Hopf boundary maximum principle violation for semilinear elliptic equations

We consider elliptic equations with non-Lipschitz nonlinearity −Δu=λ|u|β−1u−|u|α−1u−Δu=λ|u|β−1u−|u|α−1u in a smooth bounded domain Ω⊂RnΩ⊂Rn, with Dirichlet boundary conditions; here 0<α<β<10<α<β<1. We prove the existence of a weak nonnegative solution which does not satisfy the Hopf boundary maximum principle, provided that λλ is large enough and n>max{2,2(1+α)(1+β)/(1−α)(1−β)}n>max{2,2(1+α)(1+β)/(1−α)(1−β)}.