Advanced Search
Article Contents
Article Contents

Higher order conformally invariant equations in $ {\mathbb R}^3 $ with prescribed volume

The first author is supported by SNSF Grant No. P2BSP2-172064. The second author is partially supported by NSERC

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper we study the following conformally invariant poly-harmonic equation

    $ \Delta^mu = -u^\frac{3+2m}{3-2m}\quad\text{in }\mathbb{R}^3,\quad u>0, $

    with $ m = 2,3 $. We prove the existence of positive smooth radial solutions with prescribed volume $ \int_{\mathbb{R}^3} u^\frac{6}{3-2m}dx $. We show that the set of all possible values of the volume is a bounded interval $ (0,\Lambda^*] $ for $ m = 2 $, and it is $ (0,\infty) $ for $ m = 3 $. This is in sharp contrast to $ m = 1 $ case in which the volume $ \int_{\mathbb{R}^3} u^\frac{6}{3-2m}dx $ is a fixed value.

    Mathematics Subject Classification: Primary: 35J30, 35J91, 53A30.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] T. P. Branson, Group representations arising from Lorentz conformal geometry, J. Funct. Anal., 74 (1987), 199-291.  doi: 10.1016/0022-1236(87)90025-5.
    [2] L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.
    [3] S-Y. A. Chang and W. Chen, A note on a class of higher order conformally covariant equations, Discrete Contin. Dynam. Systems, 7 (2001), 275-281.  doi: 10.3934/dcds.2001.7.275.
    [4] Y. S. Choi and X. Xu, Nonlinear biharmonic equations with negative exponents, J. Diff. Equations, 246 (2009), 216-234.  doi: 10.1016/j.jde.2008.06.027.
    [5] T. V. Duoc and Q. A. Ngô, A note on positive radial solutions of $\Delta^2 u+u^{-q} = 0$ in $ \mathbb{R}^3$ with exactly quadratic growth at infinity, Diff. Int. Equations, 30 (2017), 917-928. 
    [6] T. V. Duoc and Q. A. Ngô, Exact growth at infinity for radial solutions of $\Delta^3u+u^{-q} = 0$ in $\mathbb{ \mathbb{R}}^3$, Preprint (2017), ftp://file.viasm.org/Web/TienAnPham-17/Preprint_1702.pdf.
    [7] A. Farina and A. Ferrero, Existence and stability properties of entire solutions to the polyharmonic equation $(-\Delta u)^m = e^u$ for any $m>1$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 495-528.  doi: 10.1016/j.anihpc.2014.11.005.
    [8] X. Feng and X. Xu, Entire solutions of an integral equation in $ \mathbb{R}^5$, ISRN Math. Anal., (2013), Art. ID 384394, 17 pp.
    [9] I. Guerra, A note on nonlinear biharmonic equations with negative exponents, J. Differential Equations, 253 (2012), 3147-3157.  doi: 10.1016/j.jde.2012.08.037.
    [10] X. Hunag and D. Ye, Conformal metrics in $\mathbb{R}^{2m}$ with constant $Q$-curvature and arbitrary volume, Calc. Var. Partial Differential Equations, 54 (2015), 3373-3384.  doi: 10.1007/s00526-015-0907-1.
    [11] A. Hyder, Conformally Euclidean metrics on $ \mathbb{R}^n$ with arbitrary total Q-curvature, Anal. PDE, 10 (2017), 635-652.  doi: 10.2140/apde.2017.10.635.
    [12] A. Hyder and J. Wei, Non-radial solutions to a biharmonic equation with negative exponent, Preprint (2018), http://www.math.ubc.ca/~ali.hyder/W/HW.pdf.
    [13] B. Lai, A new proof of I. Guerra's results concerning nonlinear biharmonic equations with negative exponents, J. Math. Anal. Appl., 418 (2014), 469-475.  doi: 10.1016/j.jmaa.2014.04.005.
    [14] Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. 
    [15] C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^n$, Comment. Math. Helv., 73 (1998), 206-231.  doi: 10.1007/s000140050052.
    [16] L. Martinazzi, Conformal metrics on $ \mathbb{R}^{2m}$ with constant Q-curvature and large volume, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 969-982.  doi: 10.1016/j.anihpc.2012.12.007.
    [17] P. J. McKenna and W. Reichel, Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry, Electron. J. Differential Equations, 37 (2003), 1-13. 
    [18] Q. A. Ngô, Classification of entire solutions of $(-\Delta)^n u+u^{4n-1}=0$ with exact linear growth at infinity in $\mathbb{R}^{2n-1}$, Proc. Amer. Math. Soc., 146 (2018), 2585-2600.  doi: 10.1090/proc/13960.
    [19] J. Wei and D. Ye, Nonradial solutions for a conformally invariant fourth order equation in $ \mathbb{R}^4$, Calc. Var. Partial Differential Equations, 32 (2008), 373-386.  doi: 10.1007/s00526-007-0145-2.
    [20] X. Xu, Exact solutions of nonlinear conformally invariant integral equations in $\mathbb{R}^3$, Adv. Math., 194 (2005), 485-503.  doi: 10.1016/j.aim.2004.07.004.
    [21] X. Xu and P. Yang, On a fourth order equation in $3$-$D$, ESAIM: Control Optim. Calc. Var., 8 (2002), 1029-1042.  doi: 10.1051/cocv:2002023.
    [22] P. Yang and M. Zhu, On the Paneitz energy on standard three sphere, ESAIM: Control Optim. Calc. Var., 10 (2004), 211-223.  doi: 10.1051/cocv:2004002.
  • 加载中

Article Metrics

HTML views(420) PDF downloads(232) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint