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

Filomena Pacella; Dora Salazar

doi:10.3934/dcdss.2014.7.793

Article Contents

2014, Volume 7, Issue 4: 793-805. Doi: 10.3934/dcdss.2014.7.793

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

Filomena Pacella^1, and
Dora Salazar^2,

1.
Dipartimento di Matematica, Università di Roma Sapienza, P.le A. Moro 2, 00185 Roma
2.
Centro de Modelamiento Matemático, UMI 2807 CNRS-UChile, Universidad de Chile, Blanco Encalada 2120, Piso 7, Santiago

Received: June 30, 2013

Revised: September 30, 2013

Published: July 2014

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Abstract

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 $\infty$.

Keywords:

Superlinear elliptic boundary value problems,
radial solutions,
asymptotic behaviour,
sign changing solutions,
annular domains.

Mathematics Subject Classification: Primary: 35J91; Secondary: 35B40.

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References

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