Conformal metrics on ${ℝ}^{2m}$ with constant Q-curvature and large volume
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 6, pp. 969-982.

We study conformal metrics ${g}_{u}={e}^{2u}{|dx|}^{2}$ on ${ℝ}^{2m}$ with constant Q-curvature ${Q}_{{g}_{u}}\equiv \left(2m-1\right)!$ (notice that $\left(2m-1\right)!$ is the Q-curvature of ${S}^{2m}$) and finite volume. When $m=3$ we show that there exists ${V}^{⁎}$ such that for any $V\in \left[{V}^{⁎},\infty \right)$ there is a conformal metric ${g}_{u}={e}^{2u}{|dx|}^{2}$ on ${ℝ}^{6}$ with ${Q}_{{g}_{u}}\equiv 5!$ and $\mathrm{vol}\left({g}_{u}\right)=V$. This is in sharp contrast with the four-dimensional case, treated by C.-S. Lin. We also prove that when m is odd and greater than 1, there is a constant ${V}_{m}>\mathrm{vol}\left({S}^{2m}\right)$ such that for every $V\in \left(0,{V}_{m}\right]$ there is a conformal metric ${g}_{u}={e}^{2u}{|dx|}^{2}$ on ${ℝ}^{2m}$ with ${Q}_{{g}_{u}}\equiv \left(2m-1\right)!$, $\mathrm{vol}\left(g\right)=V$. This extends a result of A. Chang and W.-X. Chen. When m is even we prove a similar result for conformal metrics of negative Q-curvature.

DOI: 10.1016/j.anihpc.2012.12.007
Keywords: Q-curvature, Paneitz operators, GMJS operators, Conformal geometry
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author = {Martinazzi, Luca},
title = {Conformal metrics on ${\mathbb{R}}^{2m}$ with constant {\protect\emph{Q}-curvature} and large volume},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Martinazzi, Luca. Conformal metrics on ${\mathbb{R}}^{2m}$ with constant Q-curvature and large volume. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 6, pp. 969-982. doi : 10.1016/j.anihpc.2012.12.007. http://www.numdam.org/articles/10.1016/j.anihpc.2012.12.007/

[1] Adimurthi, F. Robert, M. Struwe, Concentration phenomena for Liouvilleʼs equation in dimension 4, J. Eur. Math. Soc. 8 (2006), 171-180 | EuDML | MR | Zbl

[2] H. Brézis, F. Merle, Uniform estimates and blow-up behaviour for solutions of $-\Delta u=V\left(x\right){e}^{u}$ in two dimensions, Comm. Partial Differential Equations 16 (1991), 1223-1253 | Zbl

[3] S.-Y.A. Chang, Non-linear Elliptic Equations in Conformal Geometry, Zur. Lect. Notes Adv. Math., EMS (2004) | MR

[4] S.-Y.A. Chang, W. Chen, A note on a class of higher order conformally covariant equations, Discrete Contin. Dyn. Syst. 63 (2001), 275-281 | MR | Zbl

[5] S.-Y.A. Chang, P. Yang, On uniqueness of solutions of n-th order differential equations in conformal geometry, Math. Res. Lett. 4 (1997), 91-102 | MR | Zbl

[6] W. Chen, C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 no. 3 (1991), 615-622 | MR | Zbl

[7] O. Druet, F. Robert, Bubbling phenomena for fourth-order four-dimensional PDEs with exponential growth, Proc. Amer. Math. Soc. 3 (2006), 897-908 | MR | Zbl

[8] C. Fefferman, C.R. Graham, Q-curvature and Poincaré metrics, Math. Res. Lett. 9 (2002), 139-151 | MR | Zbl

[9] C. Fefferman, K. Hirachi, Ambient metric construction of Q-curvature in conformal and CR geometry, Math. Res. Lett. 10 (2003), 819-831 | MR | Zbl

[10] Y. Li, I. Shafrir, Blow-up analysis for solutions of $-\Delta u=V{e}^{u}$ in dimension two, Indiana Univ. Math. J. 43 (1994), 1255-1270 | MR | Zbl

[11] C.S. Lin, A classification of solutions of conformally invariant fourth order equations in ${ℝ}^{n}$, Comment. Math. Helv. 73 (1998), 206-231 | MR | Zbl

[12] A. Malchiodi, Compactness of solutions to some geometric fourth-order equations, J. Reine Angew. Math. 594 (2006), 137-174 | MR | Zbl

[13] A. Malchiodi, M. Struwe, Q-curvature flow on ${S}^{4}$, J. Differential Geom. 73 (2006), 1-44 | MR | Zbl

[14] L. Martinazzi, Classification of solutions to the higher order Liouvilleʼs equation on ${ℝ}^{2m}$, Math. Z. 263 (2009), 307-329 | MR | Zbl

[15] L. Martinazzi, Conformal metrics on ${ℝ}^{2m}$ with constant Q-curvature, Rend. Lincei. Mat. Appl. 19 (2008), 279-292 | MR | Zbl

[16] L. Martinazzi, Concentration-compactness phenomena in higher order Liouvilleʼs equation, J. Funct. Anal. 256 (2009), 3743-3771 | MR | Zbl

[17] L. Martinazzi, Quantization for the prescribed Q-curvature equation on open domains, Commun. Contemp. Math. 13 (2011), 533-551 | MR | Zbl

[18] C.B. Ndiaye, Ndiaye constant Q-curvature metrics in arbitrary dimension, J. Funct. Anal. 251 no. 1 (2007), 1-58 | MR | Zbl

[19] F. Robert, Concentration phenomena for a fourth order equation with exponential growth: The radial case, J. Differential Equations 231 (2006), 135-164 | MR | Zbl

[20] F. Robert, Quantization effects for a fourth order equation of exponential growth in dimension four, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 531-553 | MR | Zbl

[21] M. Struwe, A flow approach to Nirenbergʼs problem, Duke Math. J. 128 no. 1 (2005), 19-64 | MR | Zbl

[22] S. Wang, An example of a blow-up sequence for $-\Delta u=V\left(x\right){e}^{u}$, Differential Integral Equations 5 (1992), 1111-1114 | MR | Zbl

[23] J. Wei, X.-W. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann. 313 (1999), 207-228 | MR | Zbl

[24] J. Wei, D. Ye, Nonradial solutions for a conformally invariant fourth order equation in ${ℝ}^{4}$, preprint, 2006. | MR

[25] X.-W. Xu, Uniqueness theorems for integral equations and its application, J. Funct. Anal. 247 no. 1 (2007), 95-109 | MR | Zbl

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