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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 and $ \alpha, \beta>0 $, $ q_\gamma: = \frac{2n}{n+\gamma}
. 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 and 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. [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. [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. [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. [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. [6] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243. [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. [8] M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronomy Astrophys. Lib., 24 (1973), 229-238. [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. [10] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals(1), Math. Z., 27 (1928), 565-606.  doi: 10.1007/BF01171116. [11] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032. [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. [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. [14] D. Smets, J. B. Su and M. Willem, Non-radial ground states for the Henon equation, Commun. Contemp. Math., 4 (2002), 467-480.  doi: 10.1142/S0219199702000725. [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. [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.

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. [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. [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. [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. [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. [6] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243. [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. [8] M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronomy Astrophys. Lib., 24 (1973), 229-238. [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. [10] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals(1), Math. Z., 27 (1928), 565-606.  doi: 10.1007/BF01171116. [11] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032. [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. [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. [14] D. Smets, J. B. Su and M. Willem, Non-radial ground states for the Henon equation, Commun. Contemp. Math., 4 (2002), 467-480.  doi: 10.1142/S0219199702000725. [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. [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.
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