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

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August 2014, 7(4): 793-805. doi: 10.3934/dcdss.2014.7.793

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, Italy
2.	Centro de Modelamiento Matemático, UMI 2807 CNRS-UChile, Universidad de Chile, Blanco Encalada 2120, Piso 7, Santiago, Chile

Received July 2013 Revised October 2013 Published February 2014

Abstract
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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 $\infty$.

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

Mathematics Subject Classification: Primary: 35J91; Secondary: 35B4.

Citation: Filomena Pacella, Dora Salazar. Asymptotic behaviour of sign changing radial solutions of Lane Emden Problems in the annulus. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 793-805. doi: 10.3934/dcdss.2014.7.793

References:

[1]	Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity,, Proc. Amer. Math. Soc., 132 (2004), 1013. doi: 10.1090/S0002-9939-03-07301-5.
[2]	T. Bartsch, M. Clapp, M. Grossi and F. Pacella, Asymptotically radial solutions in expanding annular domains,, Math. Ann., 352 (2012), 485. doi: 10.1007/s00208-011-0646-3.
[3]	T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations,, Topol. Methods Nonlinear Anal., 22 (2003), 1.
[4]	F. Dickstein, F. Pacella and B. Scunzi, Sign-changing stationary solutions and blowup for the nonlinear heat equation in dimension two, preprint,, , ().
[5]	F. Gladiali, M. Grossi, F. Pacella and P. N. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus,, Calc. Var. Partial Differential Equations, 40 (2011), 295. doi: 10.1007/s00526-010-0341-3.
[6]	M. Grossi, Asymptotic behaviour of the Kazdan-Warner solution in the annulus,, J. Differential Equations, 223 (2006), 96. doi: 10.1016/j.jde.2005.08.003.
[7]	M. Grossi, C. Grumiau and F. Pacella, Lane Emden problems with large exponents and singular Liouville equations,, to appear in J. Math. Pures Appl., ().
[8]	W. M. Ni and R. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u,r)=0$,, Comm. Pure Appl. Math., 38 (1985), 67. doi: 10.1002/cpa.3160380105.

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References:

[1]	Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity,, Proc. Amer. Math. Soc., 132 (2004), 1013. doi: 10.1090/S0002-9939-03-07301-5.
[2]	T. Bartsch, M. Clapp, M. Grossi and F. Pacella, Asymptotically radial solutions in expanding annular domains,, Math. Ann., 352 (2012), 485. doi: 10.1007/s00208-011-0646-3.
[3]	T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations,, Topol. Methods Nonlinear Anal., 22 (2003), 1.
[4]	F. Dickstein, F. Pacella and B. Scunzi, Sign-changing stationary solutions and blowup for the nonlinear heat equation in dimension two, preprint,, , ().
[5]	F. Gladiali, M. Grossi, F. Pacella and P. N. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus,, Calc. Var. Partial Differential Equations, 40 (2011), 295. doi: 10.1007/s00526-010-0341-3.
[6]	M. Grossi, Asymptotic behaviour of the Kazdan-Warner solution in the annulus,, J. Differential Equations, 223 (2006), 96. doi: 10.1016/j.jde.2005.08.003.
[7]	M. Grossi, C. Grumiau and F. Pacella, Lane Emden problems with large exponents and singular Liouville equations,, to appear in J. Math. Pures Appl., ().
[8]	W. M. Ni and R. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u,r)=0$,, Comm. Pure Appl. Math., 38 (1985), 67. doi: 10.1002/cpa.3160380105.

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