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

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

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 $\infty$.
Citation: Filomena Pacella, Dora Salazar. Asymptotic behaviour of sign changing radial solutions of Lane Emden Problems in the annulus. Discrete and 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), 1013-1019. 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), 485-515. 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), 1-14.

[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), 295-317. 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), 96-111. 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), 67-108. doi: 10.1002/cpa.3160380105.

show all references

References:
[1]

Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Amer. Math. Soc., 132 (2004), 1013-1019. 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), 485-515. 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), 1-14.

[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), 295-317. 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), 96-111. 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), 67-108. doi: 10.1002/cpa.3160380105.

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