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- W2952493067 abstract "We consider the following fractional semilinear Neumann problem on a smooth bounded domain $Omegasubsetmathbb{R}^n$, $ngeq2$, $$begin{cases} (-varepsilonDelta)^{1/2}u+u=u^{p},&hbox{in}~Omega, partial_nu u=0,&hbox{on}~partialOmega, u>0,&hbox{in}~Omega, end{cases}$$ where $varepsilon>0$ and $1<p<(n+1)/(n-1)$. This is the fractional version of the semilinear Neumann problem studied by Lin--Ni--Takagi in the late 80's. The problem arises by considering steady states of the Keller--Segel model with nonlocal chemical concentration diffusion. Using the semigroup language for the extension method and variational techniques, we prove existence of nonconstant smooth solutions for small $varepsilon$, which are obtained by minimizing a suitable energy functional. In the case of large $varepsilon$ we obtain nonexistence of nonconstant solutions. It is also shown that as $varepsilonto0$ the solutions $u_varepsilon$ tend to zero in measure on $Omega$, while they form spikes in $overline{Omega}$. The regularity estimates of the fractional Neumann Laplacian that we develop here are essential for the analysis. The latter results are of independent interest." @default.
- W2952493067 created "2019-06-27" @default.
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- W2952493067 date "2014-06-28" @default.
- W2952493067 modified "2023-09-26" @default.
- W2952493067 title "Fractional semilinear Neumann problems arising from a fractional Keller--Segel model" @default.
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- W2952493067 doi "https://doi.org/10.48550/arxiv.1406.7406" @default.
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