On the existence of blowing-up solutions for a mean field equation
Annales de l'I.H.P. Analyse non linéaire, Volume 22 (2005) no. 2, pp. 227-257.
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     author = {Esposito, Pierpaolo and Grossi, Massimo and Pistoia, Angela},
     title = {On the existence of blowing-up solutions for a mean field equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {227--257},
     publisher = {Elsevier},
     volume = {22},
     number = {2},
     year = {2005},
     doi = {10.1016/j.anihpc.2004.12.001},
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     zbl = {1129.35376},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2004.12.001/}
}
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Esposito, Pierpaolo; Grossi, Massimo; Pistoia, Angela. On the existence of blowing-up solutions for a mean field equation. Annales de l'I.H.P. Analyse non linéaire, Volume 22 (2005) no. 2, pp. 227-257. doi : 10.1016/j.anihpc.2004.12.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2004.12.001/

[1] Ambrosetti A., Garcia Azorero J., Peral I., Perturbation of Δu+u (N+2)/(N-2) =0, the scalar curvature problem in R N , and related topics, J. Funct. Anal. 165 (1999) 117-149. | MR | Zbl

[2] Aubin T., Some Nonlinear Problems in Riemannian Geometry, Springer-Verlag, Berlin, 1998. | MR | Zbl

[3] Bahri A., Critical Point at Infinity in Some Variational Problems, Pitman Research Notes Math., vol. 182, Longman House, Harlow, 1989. | MR | Zbl

[4] Bandle C., Isoperimetric Inequalities and Applications, Pitman Monographs Studies Math., vol. 7, Pitman, 1980. | MR | Zbl

[5] Baraket S., Pacard F., Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. Partial Differential Equations 6 (1998) 1-38. | MR | Zbl

[6] Bebernes J., Eberly D., Mathematical Problems from Combustion Theory, Springer, Berlin, 1989. | MR | Zbl

[7] Bianchi G., Egnell H., A note on the Sobolev inequality, J. Funct. Anal. 100 (1991) 18-24. | MR | Zbl

[8] Brezis H., Merle F., Uniform estimates and blow-up behavior for solutions of -Δu=Vx0ex0exe u in two dimensions, Comm. Partial Differential Equations 16 (1991) 1223-1253. | MR | Zbl

[9] Caglioti E., Lions P.L., Marchioro C., Pulvirenti M., A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Comm. Math. Phys. 143 (1992) 501-525. | MR | Zbl

[10] Caglioti E., Lions P.L., Marchioro C., Pulvirenti M., A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Part II, Comm. Math. Phys. 174 (1995) 229-260. | MR | Zbl

[11] Chae D., Imanuvilov O., The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys. 215 (2000) 119-142. | MR | Zbl

[12] D. Chae, G. Tarantello, On planar electroweak vortices, Ann. Inst. H. Poincaré Analyse Non Linéaire, in press.

[13] Chandrasekhar S., An Introduction to the Study of Stellar Structure, Dover, New York, 1957. | MR | Zbl

[14] Chen W., Li C., Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991) 615-623. | MR | Zbl

[15] Chen C.C., Lin C.S., Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math. 56 (2003) 1667-1727. | MR | Zbl

[16] Chen C.C., Lin C.S., On the simmetry of blowup solutions to a mean field equation, Ann. Inst. H. Poincaré Analyse Non Linéaire 18 (2001) 271-296. | Numdam | MR | Zbl

[17] M. Del Pino, M. Kowalczyk, M. Musso, Singular limits in Liouville-type equation, preprint. | MR

[18] Dancer E.N., On the uniqueness of the positive solution of a singularly perturbed problem, Rocky Mountain J. Math. 25 (1995) 957-975. | MR | Zbl

[19] Ding W., Jost J., Li J., Wang G., Existence results for mean field equations, Ann. Inst. H. Poincaré Analyse Non Linéaire 16 (1999) 653-666. | Numdam | MR | Zbl

[20] K. El Mehdi, M. Grossi, Asymptotic estimates and qualitative properties of an elliptic problem in dimension two, preprint. | MR

[21] P. Esposito, Blow up solutions for a Liouville equation with singular data, preprint, 2003. | MR

[22] P. Esposito, A class of Liouville-type equations arising in Chern-Simons vortex theory: asymptotics and construction of blowing up solutions, Thesis, Roma “Tor Vergata”, 2003.

[23] Gelfand I.M., Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. 29 (1969) 295-381. | MR | Zbl

[24] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983. | MR | Zbl

[25] Grossi M., Pistoia A., On the effect of critical points of distance function in superlinear elliptic problems, Adv. Differential Equations 5 (2000) 1397-1420. | MR | Zbl

[26] Li Y.Y., On a singularly perturbed elliptic equation, Adv. Differential Equations 2 (1997) 955-980. | MR | Zbl

[27] Liouville J., Sur l’équation aud dérivées partielles 2 log λ/uv±2λa 2 =0, J. Math. 18 (1853) 71-72.

[28] Ma L., Wei J., Convergence for a Liouville equation, Comment. Math. Helv. 76 (2001) 506-514. | MR | Zbl

[29] Mizoguchi N., Suzuki T., Equations of gas combustion: S-shaped bifurcation and mushrooms, J. Differential Equations 134 (1997) 183-215. | MR | Zbl

[30] Moseley J.L., Asymptotic solutions for a Dirichlet problem with an exponential nonlinearity, SIAM J. Math. Anal. 14 (1983) 719-735. | MR | Zbl

[31] Moseley J.L., A two-dimensional Dirichlet problem with an exponential nonlinearity, SIAM J. Math. Anal. 14 (1983) 934-946. | MR | Zbl

[32] Moser J., A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71) 1077-1092. | MR | Zbl

[33] Musso M., Pistoia A., Multispike solutions for a nonlinear elliptic problem involving the critical Sobolev exponent, Indiana Univ. Math. J. 51 (2002) 541-579. | MR | Zbl

[34] Murrey J.D., Mathematical Biology, Springer, Berlin, 1989.

[35] Nagasaki K., Suzuki T., Asymptotic analysis for a two dimensional elliptic eigenvalue problem with exponentially dominated nonlinearity, Asymptotic Anal. 3 (1990) 173-188. | MR | Zbl

[36] Nolasco M., Non-topological N-vortex condensates for the self-dual Chern-Simons theory, Comm. Pure Appl. Math. 56 (2003) 1752-1780. | MR | Zbl

[37] Rey O., The role of Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990) 1-52. | MR | Zbl

[38] Suzuki T., Two dimensional Emden-Fowler equation with exponential nonlinearity, Nonlinear Diffusion Equations and their Equilibrium States 3 (1992) 493-512. | MR | Zbl

[39] Suzuki T., Global analysis for a two-dimensional eigenvalue problem with exponential nonlinearity, Ann. Inst. H. Poincaré Analyse Non Linéaire 9 (1992) 367-398. | Numdam | MR | Zbl

[40] Trudinger N.S., On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473-483. | MR | Zbl

[41] Weston V.H., On the asymptotic solution of a partial differential equation with exponential nonlinearity, SIAM J. Math. Anal. 9 (1978) 1030-1053. | MR | Zbl

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