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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

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), 1013-1019. doi: 10.1090/S0002-9939-03-07301-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

F. Dickstein, F. Pacella and B. Scunzi, Sign-changing stationary solutions and blowup for the nonlinear heat equation in dimension two, preprint,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

F. Dickstein, F. Pacella and B. Scunzi, Sign-changing stationary solutions and blowup for the nonlinear heat equation in dimension two, preprint,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

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