On partially free boundary solutions for elliptic problems with non-Lipschitz nonlinearities

Abstract

We show that the elliptic equation with a non-Lipschitz right-hand side, $-\Delta u=\lambda |u{|}^{\beta -1}u-|u{|}^{\alpha -1}u$ with $\lambda >0$ and $0<\alpha <\beta <1$, considered on a smooth star-shaped domain $\Omega$ subject to zero Dirichlet boundary conditions, might possess a nonnegative ground state solution which violates Hopf's maximum principle only on a nonempty subset $\Gamma$ of the boundary $\partial \Omega$ such that $\Gamma \ne \partial \Omega$.

6 pages, 1 figure. Published in Applied Mathematics Letters