Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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 $$\lim\limits_{|x-a_i|\rightarrow 0}\frac{w(x)}{|x-a_i|^{-\frac{2}{p-1}}}=\left[\mu+\frac{2}{p-1}\left(\frac{2}{p-1}+2-N\right)\right]^{1/(p-1)}.$$
    Mathematics Subject Classification: Primary: 35J60, 35B40; Secondary: 35B09.


    \begin{equation} \\ \end{equation}
  • [1]

    D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials, J. Differential Equations, 224 (2006), 332-372.doi: 10.1016/j.jde.2005.07.010.


    F. C. Cîrstea, A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials, Memoirs of AMS, 227 (2014), vi+85 pp.


    F.C. Cîrstea and Y. Du, Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity, J. Funct. Anal., 250 (2007), 317-346.doi: 10.1016/j.jfa.2007.05.005.


    F.C. Cîrstea and Y. Du, Isolated singularities for weighted quasilinear elliptic equations, J. Funct. Anal., 259 (2010), 174-202.doi: 10.1016/j.jfa.2010.03.015.


    Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Vol I: Maximum principle and applications, World Scientific Publishing, 2006.doi: 10.1142/9789812774446.


    Y. Du and Z. M. Guo, The degenerate logistic model and a singularly mixed boundary blow-up problem, Disrete Conin. Dyn. Syst., 14 (2006), 1-29.


    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), 107-124.doi: 10.1017/S0024610701002289.


    M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equations, J. Evol. Equ., 3 (2003), 637-652.doi: 10.1007/s00028-003-0122-y.


    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), 731-749.doi: 10.1112/jlms/jdv003.


    L. Wei and Z. Feng, Isolated singularity for semilinear elliptic equations, Discrete Contin. Dyn. Syst., 35 (2015), 3239-3252.doi: 10.3934/dcds.2015.35.3239.

  • 加载中

Article Metrics

HTML views() PDF downloads(151) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint