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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. [2] L. Caffarelli, B. 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.

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##### 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. [2] L. Caffarelli, B. 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.
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