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

## TL;DR

This paper investigates elliptic equations with non-Lipschitz nonlinearities, revealing solutions that violate Hopf's maximum principle on parts of the boundary, thus expanding understanding of boundary behavior in such problems.

## Contribution

It demonstrates the existence of boundary solutions that partially violate Hopf's maximum principle for elliptic equations with non-Lipschitz nonlinearities on star-shaped domains.

## Key findings

- Existence of solutions violating Hopf's maximum principle on nonempty boundary subsets.
- Solutions are nonnegative ground states with partial boundary violations.
- The results apply to elliptic problems with specific 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 $\Gamma \neq \partial\Omega$.