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ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS TO FRACTIONAL NEUMANN PROBLEMS
Published online by Cambridge University Press: 13 September 2024
Abstract
We study the asymptotic behaviour of the least energy solutions to the following class of nonlocal Neumann problems:
where $\Omega \subset \mathbb {R}^{n}$ is a bounded domain of class $C^{1,1}$, $1<p<({n+s})/({n-s}),\,n>\max \{1, 2s \}, 0<s<1, d>0$ and $\mathcal {N}_{s}u$ is the nonlocal Neumann derivative. We show that for small $d,$ the least energy solutions $u_d$ of the above problem achieve an $L^{\infty }$-bound independent of $d.$ Using this together with suitable $L^{r}$-estimates on $u_d,$ we show that the least energy solution $u_d$ achieves a maximum on the boundary of $\Omega $ for d sufficiently small.
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- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
Communicated by Florica C. Cîrstea