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Asymptotic behavior of sign-changing radial solutions of a semilinear elliptic equation in $ \mathbb{R}^2 $ when exponent approaches $ +\infty $
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China |
In this paper, we consider a semi-linear elliptic equation in $ \mathbb{R}^2 $ with the nonlinear exponent approaching infinity. We study asymptotic behavior of sign-changing once radial solutions obtained by Bartsch-Willem in [
References:
[1] |
Ad imurthi 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] |
W. W. Ao, J. C. Wei and W. Yao,
Uniqueness and nondegeneracy of sign-changing radial solution to an almost critical elliptic problem, Adv. Differential Equations, 21 (2016), 1049-1084.
|
[3] |
T. Bartsch and M. Willem,
Infinitely many radial solutions of a semilinear elliptic problem on $\mathbb{R}^N$, Arch. Rational Mech. Anal., 124 (1993), 261-276.
doi: 10.1007/BF00953069. |
[4] |
M. Ben Ayed, K. El Mehdi and M. Grossi,
Asymptotic behavior of least energy solutions of a biharmonic equation in dimension four, Indiana Univ. Math. J., 55 (2006), 1723-1749.
doi: 10.1512/iumj.2006.55.2723. |
[5] |
H. Berestyeki and P. J. Lions,
Nonlinear scalar field equations I, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[6] |
K. El Mehdi and M. Grossi,
Asymptotic estimates and qualitative properties of an elliptic problem in dimension two, Adv. Nonlinear Stud., 4 (2004), 15-36.
doi: 10.1515/ans-2004-0102. |
[7] |
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. |
[8] |
M. Grossi, C. Grumiau and F. Pacella,
Lane Emden problems with large exponents and singular Liouville equations, Indiana Univ. Math. J., 101 (2014), 735-754.
doi: 10.1016/j.matpur.2013.06.011. |
[9] |
I. Ianni and A. Saldana, Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems, https://arxiv.org/abs/1908.10503. |
[10] |
F. Pacella and D. Salazar,
Asymptotic behaviour of sign changing radial solutions of Lane Emden problems in the annulus, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 793-805.
doi: 10.3934/dcdss.2014.7.793. |
[11] |
X. B. Pan and X. F. Wang,
Blow-up behavior of ground states of semilinear elliptic equations in Rn involving critical Sobolev exponents, J. Differential Equations, 9 (1992), 78-107.
doi: 10.1016/0022-0396(92)90136-B. |
[12] |
X. F. Ren and J. C. Wei,
On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc., 343 (1994), 749-763.
doi: 10.1090/S0002-9947-1994-1232190-7. |
[13] |
X. F. Ren and J. C. Wei,
On a semilinear elliptic equation in $\mathbb{R}^2$ when the exponnet approaches infinity, J. Math. Anal. Appl., 189 (1995), 179-193.
doi: 10.1006/jmaa.1995.1011. |
[14] |
S. Santra and J. C. Wei,
Asymptotic behavior of solutions of a biharmonic Dirichlet problem with large exponents, J. Anal. Math., 115 (2011), 1-31.
doi: 10.1007/s11854-011-0021-z. |
[15] |
S. Tanaka,
Uniqueness of sign-changing radial solutions for $\Delta u-u+\left\vert{u}\right\vert^{p-1}u = 0$ in some ball and annulus, J. Math. Anal. Appl., 439 (2016), 154-170.
doi: 10.1016/j.jmaa.2016.02.036. |
[16] |
M. Willem, Mninmax Theorem, Birkhäuser, Berlin, 1996.
doi: 10.1007/978-1-4612-4146-1. |
show all references
References:
[1] |
Ad imurthi 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] |
W. W. Ao, J. C. Wei and W. Yao,
Uniqueness and nondegeneracy of sign-changing radial solution to an almost critical elliptic problem, Adv. Differential Equations, 21 (2016), 1049-1084.
|
[3] |
T. Bartsch and M. Willem,
Infinitely many radial solutions of a semilinear elliptic problem on $\mathbb{R}^N$, Arch. Rational Mech. Anal., 124 (1993), 261-276.
doi: 10.1007/BF00953069. |
[4] |
M. Ben Ayed, K. El Mehdi and M. Grossi,
Asymptotic behavior of least energy solutions of a biharmonic equation in dimension four, Indiana Univ. Math. J., 55 (2006), 1723-1749.
doi: 10.1512/iumj.2006.55.2723. |
[5] |
H. Berestyeki and P. J. Lions,
Nonlinear scalar field equations I, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[6] |
K. El Mehdi and M. Grossi,
Asymptotic estimates and qualitative properties of an elliptic problem in dimension two, Adv. Nonlinear Stud., 4 (2004), 15-36.
doi: 10.1515/ans-2004-0102. |
[7] |
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. |
[8] |
M. Grossi, C. Grumiau and F. Pacella,
Lane Emden problems with large exponents and singular Liouville equations, Indiana Univ. Math. J., 101 (2014), 735-754.
doi: 10.1016/j.matpur.2013.06.011. |
[9] |
I. Ianni and A. Saldana, Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems, https://arxiv.org/abs/1908.10503. |
[10] |
F. Pacella and D. Salazar,
Asymptotic behaviour of sign changing radial solutions of Lane Emden problems in the annulus, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 793-805.
doi: 10.3934/dcdss.2014.7.793. |
[11] |
X. B. Pan and X. F. Wang,
Blow-up behavior of ground states of semilinear elliptic equations in Rn involving critical Sobolev exponents, J. Differential Equations, 9 (1992), 78-107.
doi: 10.1016/0022-0396(92)90136-B. |
[12] |
X. F. Ren and J. C. Wei,
On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc., 343 (1994), 749-763.
doi: 10.1090/S0002-9947-1994-1232190-7. |
[13] |
X. F. Ren and J. C. Wei,
On a semilinear elliptic equation in $\mathbb{R}^2$ when the exponnet approaches infinity, J. Math. Anal. Appl., 189 (1995), 179-193.
doi: 10.1006/jmaa.1995.1011. |
[14] |
S. Santra and J. C. Wei,
Asymptotic behavior of solutions of a biharmonic Dirichlet problem with large exponents, J. Anal. Math., 115 (2011), 1-31.
doi: 10.1007/s11854-011-0021-z. |
[15] |
S. Tanaka,
Uniqueness of sign-changing radial solutions for $\Delta u-u+\left\vert{u}\right\vert^{p-1}u = 0$ in some ball and annulus, J. Math. Anal. Appl., 439 (2016), 154-170.
doi: 10.1016/j.jmaa.2016.02.036. |
[16] |
M. Willem, Mninmax Theorem, Birkhäuser, Berlin, 1996.
doi: 10.1007/978-1-4612-4146-1. |
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