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April  2017, 37(4): 1789-1818. doi: 10.3934/dcds.2017075

On a resonant mean field type equation: A "critical point at Infinity" approach

Mohameden Ahmedou 1,, , Mohamed Ben Ayed 2, and  Marcello Lucia 3,

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

* Corresponding author: Mohameden.Ahmedou@math.uni-giessen.de.

Received  February 2016 Revised  November 2016 Published  December 2016

Fund Project: This work was partially supported by a grant from the Simons Foundation (Nr. 210368 to Marcello Lucia) and the grant MTM2014-52402-C3-1-P (Spain).

We consider the following mean field type equations on domains of
$\mathbb R^2$
 under Dirichlet boundary conditions:
$\left\{ \begin{array}{l} - \Delta u = \varrho \frac{{K {e^u}}}{{\int_\Omega {K {e^u}} }}\;\;\;\;\;{\rm{in}}\;\Omega ,\\\;\;\;\;u = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{on}}\;\partial \Omega ,\end{array} \right.$
where
$K$
is a smooth positive function and
$\varrho$
is a positive real parameter.
A "critical point theory at Infinity" approach of A. Bahri to the above problem is developed for the resonant case, i.e. when the parameter
$\varrho$
is a multiple of
$8 π$
. Namely, we identify the so-called "critical points at infinity" of the associated variational problem and compute their Morse indices. We then prove some Bahri-Coron type results which can be seen as a generalization of a degree formula in the non-resonant case due to C.C.Chen and C.S.[18].
Keywords: Mean field equation, critical points at infinity, infinite dimensional Morse theory, variational and topological methods.
Mathematics Subject Classification: Primary:35C60, 35J20;Secondary:35Q70.
Citation: Mohameden Ahmedou, Mohamed Ben Ayed, Marcello Lucia. On a resonant mean field type equation: A "critical point at Infinity" approach. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1789-1818. doi: 10.3934/dcds.2017075
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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