American Institute of Mathematical Sciences

Advanced
December  2016, 36(12): 7169-7189. doi: 10.3934/dcds.2016112

Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials

Lei Wei 1, , Xiyou Cheng 2, and  Zhaosheng Feng 3,

1. 

School of Mathematical and Statistics, Jiangsu Normal University, Xuzhou 221116
2. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China
3. 

School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539

Received  January 2016 Revised  March 2016 Published  October 2016

In this paper, we study the elliptic equation with a multi-singular inverse square potential $$-\Delta u=\mu\sum_{i=1}^{k}\frac{u}{|x-a_i|^2}-u^p,\ \ x\in \mathbb{R}^N\backslash\{a_i:i\in K\},$$ where $N\geq 3$, $p>1$ and $\mu>(N-2)^2/4k$. In our discussions, the domain is the entire space, and the equation contains multiple singular points. We not only demonstrate the behavior of positive solutions near each singular point $a_i$, but also obtain the behavior of positive solutions as $|x|\rightarrow \infty$. Under suitable conditions, we show that the equation has a unique positive solution $w$, which satisfies $$\lim\limits_{|x|\rightarrow\infty}\frac{w(x)}{|x|^{-\frac{2}{p-1}}}=\left[k\mu+\frac{2}{p-1}\left(\frac{2}{p-1}+2-N\right)\right]^{1/(p-1)}$$ and $$\lim\limits_{|x-a_i|\rightarrow 0}\frac{w(x)}{|x-a_i|^{-\frac{2}{p-1}}}=\left[\mu+\frac{2}{p-1}\left(\frac{2}{p-1}+2-N\right)\right]^{1/(p-1)}.$$
Keywords: Hardy potential, uniqueness., supersolution, elliptic equations, subsolution, multi-singular points.
Mathematics Subject Classification: Primary: 35J60, 35B40; Secondary: 35B0.
Citation: Lei Wei, Xiyou Cheng, Zhaosheng Feng. Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7169-7189. doi: 10.3934/dcds.2016112
References:
[1]

D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials,, J. Differential Equations, 224 (2006), 332.  doi: 10.1016/j.jde.2005.07.010.  Google Scholar

[2]

F. C. Cîrstea, A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials,, Memoirs of AMS, (2014).   Google Scholar

[3]

F.C. Cîrstea and Y. Du, Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity,, J. Funct. Anal., 250 (2007), 317.  doi: 10.1016/j.jfa.2007.05.005.  Google Scholar

[4]

F.C. Cîrstea and Y. Du, Isolated singularities for weighted quasilinear elliptic equations,, J. Funct. Anal., 259 (2010), 174.  doi: 10.1016/j.jfa.2010.03.015.  Google Scholar

[5]

Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations,, Vol I: Maximum principle and applications, (2006).  doi: 10.1142/9789812774446.  Google Scholar

[6]

Y. Du and Z. M. Guo, The degenerate logistic model and a singularly mixed boundary blow-up problem,, Disrete Conin. Dyn. Syst., 14 (2006), 1.   Google Scholar

[7]

Y. Du and L. Ma, Logistic type equations on $\mathbbR^N$ by a squeezing method involving boundary blow-up solutions,, J. London Math. Soc., 64 (2001), 107.  doi: 10.1017/S0024610701002289.  Google Scholar

[8]

M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equations,, J. Evol. Equ., 3 (2003), 637.  doi: 10.1007/s00028-003-0122-y.  Google Scholar

[9]

L. Wei and Y. Du, Exact singular behavior of positive solutions to nonlinear elliptic equations with a Hardy potential,, J. Lond. Math. Soc., 91 (2015), 731.  doi: 10.1112/jlms/jdv003.  Google Scholar

[10]

L. Wei and Z. Feng, Isolated singularity for semilinear elliptic equations,, Discrete Contin. Dyn. Syst., 35 (2015), 3239.  doi: 10.3934/dcds.2015.35.3239.  Google Scholar

show all references

References:
[1]

D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials,, J. Differential Equations, 224 (2006), 332.  doi: 10.1016/j.jde.2005.07.010.  Google Scholar

[2]

F. C. Cîrstea, A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials,, Memoirs of AMS, (2014).   Google Scholar

[3]

F.C. Cîrstea and Y. Du, Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity,, J. Funct. Anal., 250 (2007), 317.  doi: 10.1016/j.jfa.2007.05.005.  Google Scholar

[4]

F.C. Cîrstea and Y. Du, Isolated singularities for weighted quasilinear elliptic equations,, J. Funct. Anal., 259 (2010), 174.  doi: 10.1016/j.jfa.2010.03.015.  Google Scholar

[5]

Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations,, Vol I: Maximum principle and applications, (2006).  doi: 10.1142/9789812774446.  Google Scholar

[6]

Y. Du and Z. M. Guo, The degenerate logistic model and a singularly mixed boundary blow-up problem,, Disrete Conin. Dyn. Syst., 14 (2006), 1.   Google Scholar

[7]

Y. Du and L. Ma, Logistic type equations on $\mathbbR^N$ by a squeezing method involving boundary blow-up solutions,, J. London Math. Soc., 64 (2001), 107.  doi: 10.1017/S0024610701002289.  Google Scholar

[8]

M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equations,, J. Evol. Equ., 3 (2003), 637.  doi: 10.1007/s00028-003-0122-y.  Google Scholar

[9]

L. Wei and Y. Du, Exact singular behavior of positive solutions to nonlinear elliptic equations with a Hardy potential,, J. Lond. Math. Soc., 91 (2015), 731.  doi: 10.1112/jlms/jdv003.  Google Scholar

[10]

L. Wei and Z. Feng, Isolated singularity for semilinear elliptic equations,, Discrete Contin. Dyn. Syst., 35 (2015), 3239.  doi: 10.3934/dcds.2015.35.3239.  Google Scholar

[1]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274
[2]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469
[3]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075
[4]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119
[5]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272
[6]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320
[7]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318
[8]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[9]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076
[10]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384
[11]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441
[12]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[13]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[14]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
[15]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020452
[16]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436
[17]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340
[18]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342
[19]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442
[20]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences