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u>0 & \\text{in $\\Omega$.} \\end{cases} \\end{equation}\n  where $\\Omega\\subset \\mathbb{R}^2$ is a smooth bounded domain and $p>1$ is sufficiently large.\n  When $\\Omega$ is convex, our result, combined with the characterization in [22], a result in [41] and with recent uniform estimates in \\cite{Sirakov}, gives the uniqueness of the solution to \\eqref{abstr}, for $p$ large.\n  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index and uniqueness of positive solutions of the Lane-Emden problem in planar domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Filomena Pacella, Francesca De Marchis, Isabella Ianni, Massimo Grossi","submitted_at":"2018-04-10T13:18:46Z","abstract_excerpt":"We compute the Morse index of $1$-spike solutions of the semilinear elliptic problem \\begin{equation}\\label{abstr} \\tag{$\\mathcal P_p$} \\begin{cases} -\\Delta u= u^p & \\text{in $\\Omega$} \\\\ u=0 & \\text{on $\\partial\\Omega$} \\\\ u>0 & \\text{in $\\Omega$.} \\end{cases} \\end{equation}\n  where $\\Omega\\subset \\mathbb{R}^2$ is a smooth bounded domain and $p>1$ is sufficiently large.\n  When $\\Omega$ is convex, our result, combined with the characterization in [22], a result in [41] and with recent uniform estimates in \\cite{Sirakov}, gives the uniqueness of the solution to \\eqref{abstr}, for $p$ large.\n  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