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On a resonant mean field type equation: A "critical point at Infinity" approach
1. | Mathematisches Institut der Justus-Liebig-Universität Giessen, Arndtsrasse 2, D-35392 Giessen, Germany |
2. | Université de Sfax, Faculté des Sciences, Département de Mathématiques, Route de Soukra, Sfax, Tunisia |
3. | The City University of New York, CSI, Mathematics Department, 2800 Victory Boulevard, Staten Island New York 10314, USA |
$\mathbb R^2$ |
$\left\{ \begin{array}{l} - \Delta u = \varrho \frac{{K {e^u}}}{{\int_\Omega {K {e^u}} }}\;\;\;\;\;{\rm{in}}\;\Omega ,\\\;\;\;\;u = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{on}}\;\partial \Omega ,\end{array} \right.$ |
$K$ |
$\varrho$ |
$\varrho$ |
$8 π$ |
References:
[1] |
T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-13006-3. |
[2] |
A. Bahri, Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182. Longman Scientific Technical, Harlow; copublished in the United States with John Wiley Sons, Inc. , New York, 1989. |
[3] |
A. Bahri and J.-M. Coron,
On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math, 41 (1988), 253-294.
doi: 10.1002/cpa.3160410302. |
[4] |
A. Bahri and J.-M. Coron,
The scalar-curvature problem on the standard three-dimensional sphere, J. Funct. Anal., 95 (1991), 106-172.
doi: 10.1016/0022-1236(91)90026-2. |
[5] |
A. Bahri and P. Rabinowitz,
Periodic solutions of Hamiltonian systems of 3-body type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 561-649.
|
[6] |
A. Bahri,
An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimension. A celebration of John F. Nash, Jr, Duke Math. J., 81 (1996), 323-466.
doi: 10.1215/S0012-7094-96-08116-8. |
[7] |
D. Bartolucci and C. S. Lin,
Existence and Uniqueness of mean field equation on multiply connected domains at the critical parameter, Math. Ann., 359 (2014), 1-44.
doi: 10.1007/s00208-013-0990-6. |
[8] |
D. Bartolucci and F. De Marchis,
Supercritical mean field equations on convex domains and the Onsager's statistical description of two dimensional turbulence, Arch. Ration. Mech. Anal., 217 (2015), 525-570.
doi: 10.1007/s00205-014-0836-8. |
[9] |
M. Ben Ayed, Y. Chen, H. Chtioui and M. Hammami,
On the prescribed scalar curvature problem on 4-manifolds, Duke Math. J., 84 (1996), 633-677.
doi: 10.1215/S0012-7094-96-08420-3. |
[10] |
M. Ben Ayed and M. Ould Ahmedou,
Existence and multiplicity results for a fourth order mean field equation, J. Funct. Anal., 258 (2010), 3165-3194.
doi: 10.1016/j.jfa.2010.01.009. |
[11] |
H. Brézis and F. Merle,
Uniform estimates and blow-up behavior for solutions of ions of −Δu = V (x)eu in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253.
doi: 10.1080/03605309108820797. |
[12] |
E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti,
A special class of stationary flows for two dimensional Euler equations: a statistical mechanics description, Comm. Math. Phys., 143 (1992), 501-525.
doi: 10.1007/BF02099262. |
[13] |
E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti,
A special class of stationary flows for two dimensional Euler equations: a statistical mechanics description, Part Ⅱ, Comm. Math. Phys., 174 (1995), 229-260.
doi: 10.1007/BF02099602. |
[14] |
A. Chang and P. Yang,
Prescribing Gaussian curvature on $\mathbb{S}^2$, Acta Math., 159 (1987), 215-259.
doi: 10.1007/BF02392560. |
[15] |
A. Chang and P. Yang,
Conformal deformations of metrics on $\mathbb{S}^2$, J. Diff. Geom., 27 (1988), 259-296.
|
[16] |
A. Chang, C. C. Chen and C. S. Lin,
Extremal function of a mean field equation in two dimension, in Lectures on Partial Differential Equations, New Stud. Adv. Math, 2 (2003), 61-93.
|
[17] |
C. C. Chen and C. S. Lin,
Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math., 55 (2002), 728-771.
doi: 10.1002/cpa.3014. |
[18] |
C. C. Chen and C. S. Lin,
Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math., 56 (2003), 1667-1727.
doi: 10.1002/cpa.10107. |
[19] |
F. De Marchis,
Multiplicity result for a scalar field equation on compact surfaces, Comm. PDE, 33 (2008), 2208-2224.
doi: 10.1080/03605300802523446. |
[20] |
F. De Marchis,
Generic multiplicity for a scalar field equation on compact surfaces, J. Funct. Anal., 259 (2010), 2165-2192.
doi: 10.1016/j.jfa.2010.07.003. |
[21] |
F. De Marchis,
Multiplicity of solutions for a mean field equation on compact surfaces, Boll. Unione Mat. Ital., 4 (2011), 245-257.
|
[22] |
W. Ding, J. Jost, J. Li and G. Wang,
Existence results for mean field equations, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 16 (1999), 653-666.
doi: 10.1016/S0294-1449(99)80031-6. |
[23] |
Z. Djadli,
Existence result for the mean field problem on Riemann surfaces of all genus, Commun. Contemp. Math., 10 (2008), 205-220.
doi: 10.1142/S0219199708002776. |
[24] |
Z. Djadli and A. Malchiodi,
Existence of conformal metrics with constant Q-curvature, Ann. of Math., 168 (2008), 813-858.
doi: 10.4007/annals.2008.168.813. |
[25] |
P. Esposito, M. Grossi and A. Pistoia,
On the existence of blowing up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Lineéaire, 22 (2005), 227-257.
doi: 10.1016/j.anihpc.2004.12.001. |
[26] |
Z. C. Han,
Prescribing Gaussian curvature on S2, Duke Math. J., 61 (1990), 679-703.
doi: 10.1215/S0012-7094-90-06125-3. |
[27] |
Z. C. Han,
Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. Poincaré Anal. non linéaire, 8 (1991), 159-174.
|
[28] |
M. K. H. Kiessling,
Statistical mechanics of classical particles with logarithmic interactions, Comm. Pure Appl. Math., 46 (1993), 27-56.
doi: 10.1002/cpa.3160460103. |
[29] |
Y. Y. Li and I. Shafrir,
Blow-up analysis for solutions of $-Δ u=Ve^{u}$ in dimension two, Indiana Univ. Math. J., 43 (1994), 1255-1270.
doi: 10.1512/iumj.1994.43.43054. |
[30] |
Y. Y. Li,
Harnack type inequality: The method of moving planes, Comm. Math. Phys., 200 (1999), 421-444.
doi: 10.1007/s002200050536. |
[31] |
M. Lucia,
A deformation Lemma with an application to a mean field equation, Topol. Methods Nonlinear Anal., 30 (2007), 113-138.
|
[32] |
A. Malchiodi,
Morse theory and a scalar field equation on compact surfaces, Adv. Differential Equations, 13 (2008), 1109-1129.
|
[33] |
A. Malchiodi,
Topological Methods for an elliptic equation with exponential nonlinearities, Discrete Contin. Dyn. Syst., 21 (2008), 277-294.
doi: 10.3934/dcds.2008.21.277. |
[34] |
J. Milnor, Lectures on the H-Cobordism Theorem, Princeton University Press, Princeton, 1965.
![]() |
[35] |
R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, 1994.
![]() |
[36] |
M. Struwe and G. Tarantello,
On multivortex solutions in Chern-Simons gauge theory, Boll. Unione. Math. Ital. Sez. B Artic. Ric. Mat.(8), 1 (1998), 109-121.
|
[37] |
G. Tarantello, Selfdual Gauge Field Vortices: An Analytic Approach, Progress in Nonlinear differential equations, 72, Birkhäuser Boston, Inc. Boston, MA, 2008.
doi: 10.1007/978-0-8176-4608-0. |
[38] |
Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer monographs in Mathematics, Springer Verlag, New York, Inc, 2001.
doi: 10.1007/978-1-4757-6548-9. |
[39] |
L. Zhang,
Blow up solutions of some nonlinear elliptic equation involving exponential nonlinearities, Com. Math. Phys., 268 (2006), 105-133.
doi: 10.1007/s00220-006-0092-3. |
show all references
References:
[1] |
T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-13006-3. |
[2] |
A. Bahri, Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182. Longman Scientific Technical, Harlow; copublished in the United States with John Wiley Sons, Inc. , New York, 1989. |
[3] |
A. Bahri and J.-M. Coron,
On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math, 41 (1988), 253-294.
doi: 10.1002/cpa.3160410302. |
[4] |
A. Bahri and J.-M. Coron,
The scalar-curvature problem on the standard three-dimensional sphere, J. Funct. Anal., 95 (1991), 106-172.
doi: 10.1016/0022-1236(91)90026-2. |
[5] |
A. Bahri and P. Rabinowitz,
Periodic solutions of Hamiltonian systems of 3-body type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 561-649.
|
[6] |
A. Bahri,
An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimension. A celebration of John F. Nash, Jr, Duke Math. J., 81 (1996), 323-466.
doi: 10.1215/S0012-7094-96-08116-8. |
[7] |
D. Bartolucci and C. S. Lin,
Existence and Uniqueness of mean field equation on multiply connected domains at the critical parameter, Math. Ann., 359 (2014), 1-44.
doi: 10.1007/s00208-013-0990-6. |
[8] |
D. Bartolucci and F. De Marchis,
Supercritical mean field equations on convex domains and the Onsager's statistical description of two dimensional turbulence, Arch. Ration. Mech. Anal., 217 (2015), 525-570.
doi: 10.1007/s00205-014-0836-8. |
[9] |
M. Ben Ayed, Y. Chen, H. Chtioui and M. Hammami,
On the prescribed scalar curvature problem on 4-manifolds, Duke Math. J., 84 (1996), 633-677.
doi: 10.1215/S0012-7094-96-08420-3. |
[10] |
M. Ben Ayed and M. Ould Ahmedou,
Existence and multiplicity results for a fourth order mean field equation, J. Funct. Anal., 258 (2010), 3165-3194.
doi: 10.1016/j.jfa.2010.01.009. |
[11] |
H. Brézis and F. Merle,
Uniform estimates and blow-up behavior for solutions of ions of −Δu = V (x)eu in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253.
doi: 10.1080/03605309108820797. |
[12] |
E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti,
A special class of stationary flows for two dimensional Euler equations: a statistical mechanics description, Comm. Math. Phys., 143 (1992), 501-525.
doi: 10.1007/BF02099262. |
[13] |
E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti,
A special class of stationary flows for two dimensional Euler equations: a statistical mechanics description, Part Ⅱ, Comm. Math. Phys., 174 (1995), 229-260.
doi: 10.1007/BF02099602. |
[14] |
A. Chang and P. Yang,
Prescribing Gaussian curvature on $\mathbb{S}^2$, Acta Math., 159 (1987), 215-259.
doi: 10.1007/BF02392560. |
[15] |
A. Chang and P. Yang,
Conformal deformations of metrics on $\mathbb{S}^2$, J. Diff. Geom., 27 (1988), 259-296.
|
[16] |
A. Chang, C. C. Chen and C. S. Lin,
Extremal function of a mean field equation in two dimension, in Lectures on Partial Differential Equations, New Stud. Adv. Math, 2 (2003), 61-93.
|
[17] |
C. C. Chen and C. S. Lin,
Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math., 55 (2002), 728-771.
doi: 10.1002/cpa.3014. |
[18] |
C. C. Chen and C. S. Lin,
Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math., 56 (2003), 1667-1727.
doi: 10.1002/cpa.10107. |
[19] |
F. De Marchis,
Multiplicity result for a scalar field equation on compact surfaces, Comm. PDE, 33 (2008), 2208-2224.
doi: 10.1080/03605300802523446. |
[20] |
F. De Marchis,
Generic multiplicity for a scalar field equation on compact surfaces, J. Funct. Anal., 259 (2010), 2165-2192.
doi: 10.1016/j.jfa.2010.07.003. |
[21] |
F. De Marchis,
Multiplicity of solutions for a mean field equation on compact surfaces, Boll. Unione Mat. Ital., 4 (2011), 245-257.
|
[22] |
W. Ding, J. Jost, J. Li and G. Wang,
Existence results for mean field equations, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 16 (1999), 653-666.
doi: 10.1016/S0294-1449(99)80031-6. |
[23] |
Z. Djadli,
Existence result for the mean field problem on Riemann surfaces of all genus, Commun. Contemp. Math., 10 (2008), 205-220.
doi: 10.1142/S0219199708002776. |
[24] |
Z. Djadli and A. Malchiodi,
Existence of conformal metrics with constant Q-curvature, Ann. of Math., 168 (2008), 813-858.
doi: 10.4007/annals.2008.168.813. |
[25] |
P. Esposito, M. Grossi and A. Pistoia,
On the existence of blowing up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Lineéaire, 22 (2005), 227-257.
doi: 10.1016/j.anihpc.2004.12.001. |
[26] |
Z. C. Han,
Prescribing Gaussian curvature on S2, Duke Math. J., 61 (1990), 679-703.
doi: 10.1215/S0012-7094-90-06125-3. |
[27] |
Z. C. Han,
Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. Poincaré Anal. non linéaire, 8 (1991), 159-174.
|
[28] |
M. K. H. Kiessling,
Statistical mechanics of classical particles with logarithmic interactions, Comm. Pure Appl. Math., 46 (1993), 27-56.
doi: 10.1002/cpa.3160460103. |
[29] |
Y. Y. Li and I. Shafrir,
Blow-up analysis for solutions of $-Δ u=Ve^{u}$ in dimension two, Indiana Univ. Math. J., 43 (1994), 1255-1270.
doi: 10.1512/iumj.1994.43.43054. |
[30] |
Y. Y. Li,
Harnack type inequality: The method of moving planes, Comm. Math. Phys., 200 (1999), 421-444.
doi: 10.1007/s002200050536. |
[31] |
M. Lucia,
A deformation Lemma with an application to a mean field equation, Topol. Methods Nonlinear Anal., 30 (2007), 113-138.
|
[32] |
A. Malchiodi,
Morse theory and a scalar field equation on compact surfaces, Adv. Differential Equations, 13 (2008), 1109-1129.
|
[33] |
A. Malchiodi,
Topological Methods for an elliptic equation with exponential nonlinearities, Discrete Contin. Dyn. Syst., 21 (2008), 277-294.
doi: 10.3934/dcds.2008.21.277. |
[34] |
J. Milnor, Lectures on the H-Cobordism Theorem, Princeton University Press, Princeton, 1965.
![]() |
[35] |
R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, 1994.
![]() |
[36] |
M. Struwe and G. Tarantello,
On multivortex solutions in Chern-Simons gauge theory, Boll. Unione. Math. Ital. Sez. B Artic. Ric. Mat.(8), 1 (1998), 109-121.
|
[37] |
G. Tarantello, Selfdual Gauge Field Vortices: An Analytic Approach, Progress in Nonlinear differential equations, 72, Birkhäuser Boston, Inc. Boston, MA, 2008.
doi: 10.1007/978-0-8176-4608-0. |
[38] |
Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer monographs in Mathematics, Springer Verlag, New York, Inc, 2001.
doi: 10.1007/978-1-4757-6548-9. |
[39] |
L. Zhang,
Blow up solutions of some nonlinear elliptic equation involving exponential nonlinearities, Com. Math. Phys., 268 (2006), 105-133.
doi: 10.1007/s00220-006-0092-3. |
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