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From compact semi-toric systems to Hamiltonian $S^1$-spaces
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 |
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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