• Previous Article
    Monotonicity with respect to $ p $ of the First Nontrivial Eigenvalue of the $ p $-Laplacian with Homogeneous Neumann Boundary Conditions
  • CPAA Home
  • This Issue
  • Next Article
    Existence of monotone positive solutions of a neighbour based chemotaxis model and aggregation phenomenon
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 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. GidasW. 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. 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.

[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. GidasW. 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. 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.

[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.

[1]

Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951

[2]

Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027

[3]

Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018

[4]

Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791

[5]

Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935

[6]

Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164

[7]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987

[8]

Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171

[9]

Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure and Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015

[10]

Minbo Yang, Fukun Zhao, Shunneng Zhao. Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5209-5241. doi: 10.3934/dcds.2021074

[11]

Lorenzo D'Ambrosio, Enzo Mitidieri. Hardy-Littlewood-Sobolev systems and related Liouville theorems. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 653-671. doi: 10.3934/dcdss.2014.7.653

[12]

Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057

[13]

Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1319-1345. doi: 10.3934/cpaa.2021022

[14]

Qiong Chen, Chunlai Mu, Zhaoyin Xiang. Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system. Communications on Pure and Applied Analysis, 2006, 5 (3) : 435-446. doi: 10.3934/cpaa.2006.5.435

[15]

Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure and Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018

[16]

Hua Jin, Wenbin Liu, Huixing Zhang, Jianjun Zhang. Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth. Communications on Pure and Applied Analysis, 2020, 19 (1) : 123-144. doi: 10.3934/cpaa.2020008

[17]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

[18]

Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089

[19]

Mohamed Jleli, Bessem Samet. Instantaneous blow-up for nonlinear Sobolev type equations with potentials on Riemannian manifolds. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2065-2078. doi: 10.3934/cpaa.2022036

[20]

Frank Merle, Hatem Zaag. O.D.E. type behavior of blow-up solutions of nonlinear heat equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 435-450. doi: 10.3934/dcds.2002.8.435

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (195)
  • HTML views (88)
  • Cited by (0)

Other articles
by authors

[Back to Top]