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Stinga","submitted_at":"2014-06-28T14:23:51Z","abstract_excerpt":"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. 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Volzone, P. R. Stinga","submitted_at":"2014-06-28T14:23:51Z","abstract_excerpt":"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. 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