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arXiv · 1812.08018

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

Vladimir Bobkov·Pavel Drábek·Yavdat Ilyasov
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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 $\Gamma \neq \partial\Omega$.

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Vladimir Bobkov, Pavel Drábek, Yavdat Ilyasov. 2018-12-19. On partially free boundary solutions for elliptic problems with non-Lipschitz nonlinearities. https://doi.org/10.1016/j.aml.2019.03.019

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