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

Abstract

We consider elliptic equations with non-Lipschitz nonlinearity $$ -\Delta u = \lambda |u|^{\beta-1}u-|u|^{\alpha-1}u$$ in a smooth bounded domain $\Omega \subset \mathbb{R}^n$, $n\geq 3$, with Dirichlet boundary conditions; here $0<\alpha<\beta<1$. We prove the existence of a weak nonnegative solution which does not satisfy the Hopf boundary maximum principle, provided that $\lambda$ is large enough and $n>2(1+\alpha) (1+\beta)/(1-\alpha)(1-\beta)$.