Hopf maximum principle violation for elliptic equations with non-Lipschitz nonlinearity

We consider elliptic equations with non-Lipschitz nonlinearity $-Δu = λ|u|^{β-1}u-|u|^{α-1}u$ in a smooth bounded domain $Ω\subset \mathbb{R}^n$, $n\geq 3$, 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 $n>2(1+α) (1+β)/(1-α)(1-β)$.

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