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Asymptotic behaviour of sign changing radial solutions of Lane Emden Problems in the annulus

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  • We study the asymptotic behaviour as $p\rightarrow \infty$ of the nodal radial solutions $u_p$ of the problem \begin{equation*} \left\{ \begin{array}{rlll} -\Delta u&=&|u|^{p-1}u& \text{in }\Omega \\ u&=&0& \text{on }\partial\Omega, \end{array} \right. \end{equation*} where $\Omega$ is an annulus in $\mathbb{R}^N$, $N\geq 2$. We also analyze the spectrum of the linearized operator associated to $u_p$ in the case when $u_p$ has only two nodal regions. In particular, we prove that the Morse index of $u_p$ tends to $\infty$ as $p$ goes to $\infty$.
    Mathematics Subject Classification: Primary: 35J91; Secondary: 35B40.


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