Conformal metrics on \mathbb{R}^{2m} with constant Q-curvature, prescribed volume and asymptotic behavior

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January 2015, 35(1): 283-299. doi: 10.3934/dcds.2015.35.283

Conformal metrics on $\mathbb{R}^{2m}$ with constant Q-curvature, prescribed volume and asymptotic behavior

Ali Hyder ^1, and Luca Martinazzi ^1,

1.	University of Basel, Department of Mathematics and Computer Science, Rheinsprung 21, 4051 Basel, Switzerland, Switzerland

Received January 2014 Revised May 2014 Published August 2014

Abstract
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We study the solutions $u\in C^\infty(\mathbb{R}^{2m})$ of the problem \begin{equation}\label{P0} (-\Delta)^mu=\bar Qe^{2mu}, \text{ where }\bar Q=\pm (2m-1)!, \quad V :=\int_{\mathbb{R}^{2m}}e^{2mu}dx <\infty,(1) \end{equation} particularly when $m>1$. Problem (1) corresponds to finding conformal metrics $g_u:=e^{2u}|dx|^2$ on $\mathbb{R}^{2m}$ with constant $Q$-curvature $\bar Q$ and finite volume $V$. Extending previous works of Chang-Chen, and Wei-Ye, we show that both the value $V$ and the asymptotic behavior of $u(x)$ as $|x|\to \infty$ can be simultaneously prescribed, under certain restrictions. When $\bar Q= (2m-1)!$ we need to assume $V < vol(S^{2m})$, but surprisingly for $\bar Q=-(2m-1)!$ the volume $V$ can be chosen arbitrarily.

Keywords: semilinear elliptic equations, conformal geometry, GJMS operators., $Q$-curvature.

Mathematics Subject Classification: Primary: 35J30, 35J60, 53A3.

Citation: Ali Hyder, Luca Martinazzi. Conformal metrics on $\mathbb{R}^{2m}$ with constant Q-curvature, prescribed volume and asymptotic behavior. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 283-299. doi: 10.3934/dcds.2015.35.283

References:

[1]	H. Brézis and F. Merle, Uniform estimates and blow-up behavior for solutions of $-\Delta u=V(x)e^u$ in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253. doi: 10.1080/03605309108820797.
[2]	Sun-Yung 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.
[3]	Sun-Yung A. Chang and P. Yang, On uniqueness of solutions of $n$-th order differential equations in conformal geometry, Math. Res. Lett., 4 (1997), 91-102. doi: 10.4310/MRL.1997.v4.n1.a9.
[4]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.
[5]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
[6]	E. A. Gorin, Asymptotic properties of polynomials and algebraic functions of several variables, Russ. Math. Surv., 16 (1961), 91-118.
[7]	C. R. Graham, R. Jenne, L. Mason and G. Sparling, Conformally invariant powers of the Laplacian. I. existence, J. London Math. Soc., 46 (1992), 557-565. doi: 10.1112/jlms/s2-46.3.557.
[8]	T. Jin, A. Maalaoui, L. Martinazzi and J. Xiong, Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three, to appear in Calc. Var. Partial Differential Equations, (2014). doi: 10.1007/s00526-014-0718-9.
[9]	C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^n$, Comment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.
[10]	R. C. McOwen, The behavior of the Laplacian on weighted Sobolev spaces, Comm. Pure Appl. Math., 32 (1979), 783-795. doi: 10.1002/cpa.3160320604.
[11]	L. Martinazzi, Conformal metrics on $\mathbbR^{2m}$ with constant $Q$-curvature, Rend. Lincei. Mat. Appl., 19 (2008), 279-292. doi: 10.4171/RLM/525.
[12]	L. Martinazzi, Classification of solutions to the higher order Liouville's equation on $\mathbbR^{2m}$, Math. Z., 263, (2009), 307-329. doi: 10.1007/s00209-008-0419-1.
[13]	L. Martinazzi, Quantization for the prescribed Q-curvature equation on open domains, Commun. Contemp. Math., 13 (2011), 533-551. doi: 10.1142/S0219199711004373.
[14]	L. Martinazzi, Conformal metrics on $\mathbbR^{2m}$ with constant Q-curvature and large volume, Ann. Inst. Henri Poincaré (C), 30 (2013), 969-982. doi: 10.1016/j.anihpc.2012.12.007.
[15]	L. Martinazzi and M. Petrache, Asymptotics and quantization for a mean-field equation of higher order, Comm. Partial Differential Equations, 35 (2010), 443-464. doi: 10.1080/03605300903296330.
[16]	F. Robert, Quantization effects for a fourth order equation of exponential growth in dimension four, Proc. Roy. Soc. Edinburgh Sec. A, 137 (2007), 531-553. doi: 10.1017/S0308210506000096.
[17]	J. Wei and D. Ye, Nonradial solutions for a conformally invariant fourth order equation in $\mathbbR^4$, Calc. Var. Partial Differential Equations, 32 (2008), 373-386. doi: 10.1007/s00526-007-0145-2.

show all references

References:

[1]	H. Brézis and F. Merle, Uniform estimates and blow-up behavior for solutions of $-\Delta u=V(x)e^u$ in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253. doi: 10.1080/03605309108820797.
[2]	Sun-Yung 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.
[3]	Sun-Yung A. Chang and P. Yang, On uniqueness of solutions of $n$-th order differential equations in conformal geometry, Math. Res. Lett., 4 (1997), 91-102. doi: 10.4310/MRL.1997.v4.n1.a9.
[4]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.
[5]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
[6]	E. A. Gorin, Asymptotic properties of polynomials and algebraic functions of several variables, Russ. Math. Surv., 16 (1961), 91-118.
[7]	C. R. Graham, R. Jenne, L. Mason and G. Sparling, Conformally invariant powers of the Laplacian. I. existence, J. London Math. Soc., 46 (1992), 557-565. doi: 10.1112/jlms/s2-46.3.557.
[8]	T. Jin, A. Maalaoui, L. Martinazzi and J. Xiong, Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three, to appear in Calc. Var. Partial Differential Equations, (2014). doi: 10.1007/s00526-014-0718-9.
[9]	C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^n$, Comment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.
[10]	R. C. McOwen, The behavior of the Laplacian on weighted Sobolev spaces, Comm. Pure Appl. Math., 32 (1979), 783-795. doi: 10.1002/cpa.3160320604.
[11]	L. Martinazzi, Conformal metrics on $\mathbbR^{2m}$ with constant $Q$-curvature, Rend. Lincei. Mat. Appl., 19 (2008), 279-292. doi: 10.4171/RLM/525.
[12]	L. Martinazzi, Classification of solutions to the higher order Liouville's equation on $\mathbbR^{2m}$, Math. Z., 263, (2009), 307-329. doi: 10.1007/s00209-008-0419-1.
[13]	L. Martinazzi, Quantization for the prescribed Q-curvature equation on open domains, Commun. Contemp. Math., 13 (2011), 533-551. doi: 10.1142/S0219199711004373.
[14]	L. Martinazzi, Conformal metrics on $\mathbbR^{2m}$ with constant Q-curvature and large volume, Ann. Inst. Henri Poincaré (C), 30 (2013), 969-982. doi: 10.1016/j.anihpc.2012.12.007.
[15]	L. Martinazzi and M. Petrache, Asymptotics and quantization for a mean-field equation of higher order, Comm. Partial Differential Equations, 35 (2010), 443-464. doi: 10.1080/03605300903296330.
[16]	F. Robert, Quantization effects for a fourth order equation of exponential growth in dimension four, Proc. Roy. Soc. Edinburgh Sec. A, 137 (2007), 531-553. doi: 10.1017/S0308210506000096.
[17]	J. Wei and D. Ye, Nonradial solutions for a conformally invariant fourth order equation in $\mathbbR^4$, Calc. Var. Partial Differential Equations, 32 (2008), 373-386. doi: 10.1007/s00526-007-0145-2.

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