Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fractional semilinear Neumann problems arising from a fractional
  Keller--Segel model   We consider the following fractional semilinear Neumann problem on a smooth
bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, $$\begin{cases}
(-\varepsilon\Delta)^{1/2}u+u=u^{p},&\hbox{in}~\Omega,\\ \partial_\nu
u=0,&\hbox{on}~\partial\Omega,\\ u>0,&\hbox{in}~\Omega, \end{cases}$$ 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
$\varepsilon\to0$ 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.

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The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: physics
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The categories of this paper are: ao-sci