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September  2020, 19(9): 4349-4362. doi: 10.3934/cpaa.2020196

Asymptotic behavior of solutions for nonlinear integral equations with Hénon type on the unit Ball

Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China

* Corresponding author

Received  September 2019 Revised  March 2020 Published  June 2020

Fund Project: The corresponding author is supported by the National Natural Sciences Foundations of China(No:11671128)

In this paper, we consider the problem
$ \begin{equation*} f^{q-1}(x) = \int_{\Omega}\frac{|x|^{\alpha}|y|^{\beta}f(y)}{|x-y|^{n-\gamma}}dy, \; f>0, \; x\in\overline{\Omega}, \end{equation*} $
where
$ \Omega $
is the unit ball in
$ \mathbb{R}^n(n\geq3) $
centered at the origin,
$ 1<\gamma<n $
and
$ \alpha, \beta>0 $
,
$ q_\gamma: = \frac{2n}{n+\gamma}<q<2 $
. We will investigate the asymptotic behavior of energy maximizing positive solution as
$ q\rightarrow (\frac{2n}{n+\gamma})^{+} = (q_\gamma)^+ $
. We also show that the energy maximizing positive solution concentrate at a point, which is located at the boundary as
$ q\rightarrow (q_\gamma)^{+} $
. In addition, the energy maximizing positive solution is non-radial provided that
$ q $
closes to
$ q_\gamma $
.
Citation: Ziyi Cai, Haiyang He. Asymptotic behavior of solutions for nonlinear integral equations with Hénon type on the unit Ball. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4349-4362. doi: 10.3934/cpaa.2020196
References:
[1]

J. Byeon and Z. Q. Wang, On the Hénon equation: Asymptotic profile of ground state I, Ann. Inst. Henri Poincare, 23 (2006), 803-828.  doi: 10.1016/j.anihpc.2006.04.001.  Google Scholar

[2]

J. Byeon and Z. Q. Wang, On the Hénon equation: Asymptotic profile of ground state II, J. Differ. Equ., 216 (2005), 78-108.  doi: 10.1016/j.jde.2005.02.018.  Google Scholar

[3]

Daomin Cao and Shuangjie Peng, The asymptotic behaviour of the ground state solution for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1-17.  doi: 10.1016/S0022-247X(02)00292-5.  Google Scholar

[4]

W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962.  doi: 10.1090/S0002-9939-07-09232-5.  Google Scholar

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J. Dou and M. Zhu, Nonlinear integral equations on bounded domains, J. Funct. Anal., 277 (2019), 111-134.  doi: 10.1016/j.jfa.2018.05.020.  Google Scholar

[6]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.   Google Scholar

[7]

Q. Q. Guo, Blow up analysis for integral equations on bouned domain, J. Differ. Equ., 266 (2019), 8258-8280.  doi: 10.1016/j.jde.2018.12.028.  Google Scholar

[8]

M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronomy Astrophys. Lib., 24 (1973), 229-238.   Google Scholar

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G. H. Hardy and J. E. Littlewood, On certain inequalities connected with the calculus of varations, J. Lond. Math. Soc., 5 (1930), 34-39.  doi: 10.1112/jlms/s1-5.1.34.  Google Scholar

[10]

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals(1), Math. Z., 27 (1928), 565-606.  doi: 10.1007/BF01171116.  Google Scholar

[11]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

[12]

W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807.  doi: 10.1512/iumj.1982.31.31056.  Google Scholar

[13]

D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problem, Calc. Var. Partial Differ. Equ., 18 (2003), 57-75.  doi: 10.1007/s00526-002-0180-y.  Google Scholar

[14]

D. SmetsJ. B. Su and M. Willem, Non-radial ground states for the Henon equation, Commun. Contemp. Math., 4 (2002), 467-480.  doi: 10.1142/S0219199702000725.  Google Scholar

[15]

S. L. Sobolev, On a theorem of functional analysis, Mat. Sb. (N. S.), 4 (1938), 471–479, Amer. Math. Soc. Transl. Ser., 34(1963), 39-68. Google Scholar

[16]

S. T. Zhang and Y. Z. Han, Extremal problem of Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds, preprint, arXiv: 1901.02309. doi: 10.1016/j.jde.2015.06.032.  Google Scholar

show all references

References:
[1]

J. Byeon and Z. Q. Wang, On the Hénon equation: Asymptotic profile of ground state I, Ann. Inst. Henri Poincare, 23 (2006), 803-828.  doi: 10.1016/j.anihpc.2006.04.001.  Google Scholar

[2]

J. Byeon and Z. Q. Wang, On the Hénon equation: Asymptotic profile of ground state II, J. Differ. Equ., 216 (2005), 78-108.  doi: 10.1016/j.jde.2005.02.018.  Google Scholar

[3]

Daomin Cao and Shuangjie Peng, The asymptotic behaviour of the ground state solution for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1-17.  doi: 10.1016/S0022-247X(02)00292-5.  Google Scholar

[4]

W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962.  doi: 10.1090/S0002-9939-07-09232-5.  Google Scholar

[5]

J. Dou and M. Zhu, Nonlinear integral equations on bounded domains, J. Funct. Anal., 277 (2019), 111-134.  doi: 10.1016/j.jfa.2018.05.020.  Google Scholar

[6]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.   Google Scholar

[7]

Q. Q. Guo, Blow up analysis for integral equations on bouned domain, J. Differ. Equ., 266 (2019), 8258-8280.  doi: 10.1016/j.jde.2018.12.028.  Google Scholar

[8]

M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronomy Astrophys. Lib., 24 (1973), 229-238.   Google Scholar

[9]

G. H. Hardy and J. E. Littlewood, On certain inequalities connected with the calculus of varations, J. Lond. Math. Soc., 5 (1930), 34-39.  doi: 10.1112/jlms/s1-5.1.34.  Google Scholar

[10]

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals(1), Math. Z., 27 (1928), 565-606.  doi: 10.1007/BF01171116.  Google Scholar

[11]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

[12]

W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807.  doi: 10.1512/iumj.1982.31.31056.  Google Scholar

[13]

D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problem, Calc. Var. Partial Differ. Equ., 18 (2003), 57-75.  doi: 10.1007/s00526-002-0180-y.  Google Scholar

[14]

D. SmetsJ. B. Su and M. Willem, Non-radial ground states for the Henon equation, Commun. Contemp. Math., 4 (2002), 467-480.  doi: 10.1142/S0219199702000725.  Google Scholar

[15]

S. L. Sobolev, On a theorem of functional analysis, Mat. Sb. (N. S.), 4 (1938), 471–479, Amer. Math. Soc. Transl. Ser., 34(1963), 39-68. Google Scholar

[16]

S. T. Zhang and Y. Z. Han, Extremal problem of Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds, preprint, arXiv: 1901.02309. doi: 10.1016/j.jde.2015.06.032.  Google Scholar

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