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A Morse index formula for the Lane-Emden problem Ianni, Isabella


We consider the semilinear Lane-Emden problem \begin{equation} \label{problemAbstract} \left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }B\\ u=0\qquad\qquad\qquad\mbox{ on }\partial B \end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $B$ is the unit ball of $\mathbb R^N$, $N\geq3$, centered at the origin and $p\in(1,p_S)$, $p_S=\frac{N+2}{N-2}$. We compute the Morse index of any radial solution $u_p$ of \eqref{problemAbstract}, for $p$ sufficiently close to $p_S$. The proof exploits the asymptotic behavior of $u_p$ as $p\rightarrow p_S$ and the analysis of a limit eigenvalue problem. The result is obtained in collaboration with F. De Marchis and F. Pacella.

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