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

Ziyi Cai; Haiyang He

doi:10.3934/cpaa.2020196

Article Contents

2020, Volume 19, Issue 9: 4349-4362. Doi: 10.3934/cpaa.2020196

This issue Previous Article Existence of monotone positive solutions of a neighbour based chemotaxis model and aggregation phenomenon Next Article Monotonicity with respect to

$ p $

of the First Nontrivial Eigenvalue of the

$ p $

-Laplacian with Homogeneous Neumann Boundary Conditions

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

Ziyi Cai and
Haiyang He

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

^* Corresponding author
^* Corresponding author

Received: August 31, 2019

Revised: February 29, 2020

Published: June 2020

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

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Abstract

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 $.

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Mathematics Subject Classification: Primary: 45G05, 47J30; Secondary: 35B44.

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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.