American Institute of Mathematical Sciences

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

 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 $$\label{P0} (-\Delta)^mu=\bar Qe^{2mu}, \text{ where }\bar Q=\pm (2m-1)!, \quad V :=\int_{\mathbb{R}^{2m}}e^{2mu}dx <\infty,(1)$$ 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.
Citation: Ali Hyder, Luca Martinazzi. Conformal metrics on $\mathbb{R}^{2m}$ with constant Q-curvature, prescribed volume and asymptotic behavior. Discrete & Continuous Dynamical Systems - A, 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. 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. 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. 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. 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, (1998). [6] E. A. Gorin, Asymptotic properties of polynomials and algebraic functions of several variables,, Russ. Math. Surv., 16 (1961), 91. [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. 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. doi: 10.1007/s000140050052. [10] R. C. McOwen, The behavior of the Laplacian on weighted Sobolev spaces,, Comm. Pure Appl. Math., 32 (1979), 783. doi: 10.1002/cpa.3160320604. [11] L. Martinazzi, Conformal metrics on $\mathbbR^{2m}$ with constant $Q$-curvature,, Rend. Lincei. Mat. Appl., 19 (2008), 279. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00526-007-0145-2.

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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. 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. 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. 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. 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, (1998). [6] E. A. Gorin, Asymptotic properties of polynomials and algebraic functions of several variables,, Russ. Math. Surv., 16 (1961), 91. [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. 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. doi: 10.1007/s000140050052. [10] R. C. McOwen, The behavior of the Laplacian on weighted Sobolev spaces,, Comm. Pure Appl. Math., 32 (1979), 783. doi: 10.1002/cpa.3160320604. [11] L. Martinazzi, Conformal metrics on $\mathbbR^{2m}$ with constant $Q$-curvature,, Rend. Lincei. Mat. Appl., 19 (2008), 279. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00526-007-0145-2.
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