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 $\Omega \subset {\mathbb{R}}^n$, $n\ge 3$, 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 $n>2(1+\alpha )(1+\beta )/(1-\alpha )(1-\beta )$.

arXiv comment15 pages

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