• Previous Article
    Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing
  • CPAA Home
  • This Issue
  • Next Article
    Scattering in the weighted $ L^2 $-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity
September  2019, 18(5): 2757-2764. doi: 10.3934/cpaa.2019123

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

Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T1Z2, Canada

Received  October 2018 Revised  January 2019 Published  April 2019

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

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.
Citation: Ali Hyder, Juncheng Wei. Higher order conformally invariant equations in $ {\mathbb R}^3 $ with prescribed volume. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2757-2764. doi: 10.3934/cpaa.2019123
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[14]

Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[14]

Y. Li, Remarks on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

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

[2]

Huyuan Chen, Dong Ye, Feng Zhou. On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3201-3214. doi: 10.3934/dcds.2020125

[3]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115

[4]

Tomasz Szostok. Inequalities of Hermite-Hadamard type for higher order convex functions, revisited. Communications on Pure & Applied Analysis, 2021, 20 (2) : 903-914. doi: 10.3934/cpaa.2020296

[5]

Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365

[6]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221

[7]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[8]

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

[9]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052

[10]

Kevin Li. Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021003

[11]

Honglei Lang, Yunhe Sheng. Linearization of the higher analogue of Courant algebroids. Journal of Geometric Mechanics, 2020, 12 (4) : 585-606. doi: 10.3934/jgm.2020025

[12]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

[13]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[14]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355

[15]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284

[16]

Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325

[17]

Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391

[18]

Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226

[19]

Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389

[20]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (102)
  • HTML views (205)
  • Cited by (0)

Other articles
by authors

[Back to Top]