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August  2020, 40(8): 5047-5077. doi: 10.3934/dcds.2020211

Asymptotic behavior of sign-changing radial solutions of a semilinear elliptic equation in $ \mathbb{R}^2 $ when exponent approaches $ +\infty $

Weiwei Ao and  Chao Liu

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received  December 2019 Published  May 2020

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 [3] and [16]. Assuming $ u_{p}(0)>0 $, we prove that a suitable rescaling of the positive part $ u^+_{p} $ converges to the unique regular solution of Liouville equation in $ \mathbb{R}^2 $, while a suitable rescaling of the negative part $ u^-_{p} $ converges to a solution of a singular Liouville equation in $ \mathbb{R}^2 $. We also obtain the asymptotic value of the $ L^\infty $-norms of $ u^-_{p} $ and $ u^+_{p} $. Moreover, we show that $ pu_p $ blow up at the origin and $ pu_p $ convergence to the fundamental solution of $ -\Delta +1 $ in $ \mathbb{R}^2 $ (up to a multiplier).

Keywords: Asymptotic behavior, blow-up analysis, Liouville equations.
Mathematics Subject Classification: Primary: 35J47; Secondary: 35J60.
Citation: Weiwei Ao, Chao Liu. Asymptotic behavior of sign-changing radial solutions of a semilinear elliptic equation in $ \mathbb{R}^2 $ when exponent approaches $ +\infty $. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 5047-5077. doi: 10.3934/dcds.2020211
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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

H. Berestyeki and P. J. Lions, Nonlinear scalar field equations I, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

I. Ianni and A. Saldana, Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems, https://arxiv.org/abs/1908.10503. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. Willem, Mninmax Theorem, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

H. Berestyeki and P. J. Lions, Nonlinear scalar field equations I, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

I. Ianni and A. Saldana, Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems, https://arxiv.org/abs/1908.10503. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. Willem, Mninmax Theorem, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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