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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. A. Gorin, Asymptotic properties of polynomials and algebraic functions of several variables, Russ. Math. Surv., 16 (1961), 91-118.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

L. Martinazzi, Quantization for the prescribed Q-curvature equation on open domains, Commun. Contemp. Math., 13 (2011), 533-551. doi: 10.1142/S0219199711004373.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. A. Gorin, Asymptotic properties of polynomials and algebraic functions of several variables, Russ. Math. Surv., 16 (1961), 91-118.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

L. Martinazzi, Quantization for the prescribed Q-curvature equation on open domains, Commun. Contemp. Math., 13 (2011), 533-551. doi: 10.1142/S0219199711004373.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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