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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, $-Δu = λ|u|^{β-1}u - |u|^{α-1}u$ with $λ>0$ and $0<α<β<1$, considered on a smooth star-shaped domain $Ω$ subject to zero Dirichlet boundary conditions, might possess a nonnegative ground state solution which violates Hopf's maximum principle only on a nonempty subset $Γ$ of the boundary $\partialΩ$ such that $Γ\neq \partialΩ$.

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Vladimir Bobkov, Pavel Drábek, Yavdat Ilyasov. 2019-04-03. 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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