# American Institute of Mathematical Sciences

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

## 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

* 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].
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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