American Institute of Mathematical Sciences

Advanced
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.  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.  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.   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.  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.  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.  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.  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.  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.   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.  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.  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.  doi: 10.1002/cpa.3160380105.  Google Scholar

[1]

Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256
[2]

Chia-Yu Hsieh. Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2657-2681. doi: 10.3934/dcdsb.2018269
[3]

M. Gaudenzi, P. Habets, F. Zanolin. Positive solutions of superlinear boundary value problems with singular indefinite weight. Communications on Pure & Applied Analysis, 2003, 2 (3) : 411-423. doi: 10.3934/cpaa.2003.2.411
[4]

Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713
[5]

Riccardo Molle, Donato Passaseo. On the behaviour of the solutions for a class of nonlinear elliptic problems in exterior domains. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 445-454. doi: 10.3934/dcds.1998.4.445
[6]

Monica Musso, A. Pistoia. Sign changing solutions to a Bahri-Coron's problem in pierced domains. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 295-306. doi: 10.3934/dcds.2008.21.295
[7]

Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439
[8]

Gabriele Cora, Alessandro Iacopetti. Sign-changing bubble-tower solutions to fractional semilinear elliptic problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6149-6173. doi: 10.3934/dcds.2019268
[9]

Maria Rosaria Lancia, Paola Vernole. The Stokes problem in fractal domains: Asymptotic behaviour of the solutions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1553-1565. doi: 10.3934/dcdss.2020088
[10]

Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151
[11]

Aixia Qian, Shujie Li. Multiple sign-changing solutions of an elliptic eigenvalue problem. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 737-746. doi: 10.3934/dcds.2005.12.737
[12]

Yuxin Ge, Monica Musso, A. Pistoia, Daniel Pollack. A refined result on sign changing solutions for a critical elliptic problem. Communications on Pure & Applied Analysis, 2013, 12 (1) : 125-155. doi: 10.3934/cpaa.2013.12.125
[13]

A. El Hamidi. Multiple solutions with changing sign energy to a nonlinear elliptic equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 253-265. doi: 10.3934/cpaa.2004.3.253
[14]

Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084
[15]

Norimichi Hirano, A. M. Micheletti, A. Pistoia. Existence of sign changing solutions for some critical problems on $\mathbb R^N$. Communications on Pure & Applied Analysis, 2005, 4 (1) : 143-164. doi: 10.3934/cpaa.2005.4.143
[16]

Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489
[17]

Guglielmo Feltrin. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Conference Publications, 2015, 2015 (special) : 436-445. doi: 10.3934/proc.2015.0436
[18]

Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41
[19]

João Marcos do Ó, Sebastián Lorca, Justino Sánchez, Pedro Ubilla. Positive radial solutions for some quasilinear elliptic systems in exterior domains. Communications on Pure & Applied Analysis, 2006, 5 (3) : 571-581. doi: 10.3934/cpaa.2006.5.571
[20]

Yuanxiao Li, Ming Mei, Kaijun Zhang. Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 883-908. doi: 10.3934/dcdsb.2016.21.883

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences