Profile of bubbling solutions to a Liouville system

Lin, Chang-Shou; Zhang, Lei

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 117-143.

En plusieurs champs de Physique, Chimie et Écologie, quelques modèles sont décrits par les systèmes de Liouville. Dans cet article nous prouvons d'abord un résultat de caractère unique pour un système de Liouville dans ${ℝ}^2$. Alors nous établissons une estimation uniforme pour les solutions d'explosion d'un système de Liouville localement défini prés d'un point d'explosion isolé. Le résultat d'unicité, aussi bien que les estimations uniformes locales sont les ingrédients cruciaux pour obtenir a priori l'estimation, les formules comptant le degré, et l'existence pour les systèmes de Liouville définis sur des surfaces de Reimann.

In several fields of Physics, Chemistry and Ecology, some models are described by Liouville systems. In this article we first prove a uniqueness result for a Liouville system in ${ℝ}^2$. Then we establish a uniform estimate for bubbling solutions of a locally defined Liouville system near an isolated blowup point. The uniqueness result, as well as the local uniform estimates are crucial ingredients for obtaining a priori estimate, degree counting formulas and existence results for Liouville systems defined on Riemann surfaces.

Classification : 35J60, 35J55
Mots clés : Liouville system, Uniqueness results for elliptic systems, A priori estimate

[1] J.J. Aly, Thermodynamics of a two-dimensional self-gravitating system, Phys. Rev. A 49 no. 5 (1994), 3771-3783

[2] W.H. Bennet, Magnetically self-focusing streams, Phys. Rev. 45 (1934), 890-897

[3] D. Bartolucci, C.C. Chen, C.S. Lin, G. Tarantello, Profile of blow-up solutions to mean field equations with singular data, Comm. Partial Differential Equations 29 no. 7–8 (2004), 1241-1265 | Zbl

[4] P. Biler, T. Nadzieja, Existence and nonexistence of solutions of a model of gravitational interactions of particles, I & II, Colloq. Math. 66 (1994), 319-334, Colloq. Math. 67 (1994), 297-309 | EuDML | Zbl

[5] H. Brezis, F. Merle, Uniform estimates and blow-up behavior for solutions of $-\Delta u=V\left(x\right)e^u$ in two dimensions, Comm. Partial Differential Equations 16 no. 8–9 (1991), 1223-1253 | Zbl

[6] L.A. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 no. 3 (1989), 271-297 | Zbl

[7] S.A. Chang, P. Yang, Conformal deformations of metrics on ${𝕊}^2$, J. Differential Geom. 27 (1988), 256-296

[8] S. Chanillo, M.K.-H. Kiessling, Conformally invariant systems of nonlinear PDE of Liouville type, Geom. Funct. Anal. 5 no. 6 (1995), 924-947 | EuDML | Zbl

[9] C.C. Chen, C.S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. 55 no. 6 (2002), 728-771 | Zbl

[10] C.C. Chen, C.S. Lin, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math. 56 no. 12 (2003), 1667-1727 | Zbl

[11] W.X. Chen, C.M. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 no. 3 (1991), 615-622 | Zbl

[12] S. Childress, J.K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci. 56 (1981), 217-237 | Zbl

[13] M. Chipot, I. Shafrir, G. Wolansky, On the solutions of Liouville systems, J. Differential Equations 140 no. 1 (1997), 59-105 | Zbl

[14] P. Debye, E. Huckel, Zur Theorie der Electrolyte, Phys. Z. 24 (1923), 305-325 | JFM

[15] G. Dunne, Self-Dual Chern–Simons Theories, Lecture Notes in Phys. vol. m36, Springer-Verlag, Berlin (1995) | Zbl

[16] J. Jost, G. Wang, Classification of solutions of a Toda system in ${ℝ}^2$, Int. Math. Res. Not. 6 (2002), 277-290 | Zbl

[17] J. Jost, G. Wang, Analytic aspects of the Toda system, I. A Moser–Trudinger inequality, Comm. Pure Appl. Math. 54 no. 11 (2001), 1289-1319 | Zbl

[18] J. Jost, C.S. Lin, G. Wang, Analytic aspects of the Toda system, II. Bubbling behavior and existence of solutions, Comm. Pure Appl. Math. 59 no. 4 (2006), 526-558 | Zbl

[19] E.F. Keller, L.A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol. 30 (1971), 235-248 | Zbl

[20] M.K.-H. Kiessling, J.L. Lebowitz, Dissipative stationary plasmas: Kinetic modeling Bennet pinch, and generalizations, Phys. Plasmas 1 (1994), 1841-1849

[21] Y.Y. Li, Harnack type inequality: The method of moving planes, Comm. Math. Phys. 200 no. 2 (1999), 421-444 | Zbl

[22] C.S. Lin, A classification of solutions of a conformally invariant fourth order equation in ${ℝ}^n$, Comment. Math. Helv. 73 no. 2 (1998), 206-231 | Zbl

[23] C.S. Lin, L. Zhang, Topological degree for some Liouville systems on Riemann surfaces, in preparation

[24] M.S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices, J. Math. Anal. Appl. 49 (1975), 215-225 | Zbl

[25] I. Rubinstein, Electro-Diffusion of Ions, SIAM Stud. Appl. Math. vol. 11, SIAM, Philadelphia, PA (1990)

[26] G. Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations, Arch. Ration. Mech. Anal. 119 (1992), 355-391 | Zbl

[27] G. Wolansky, On the evolution of self-interacting clusters and applications to semi-linear equations with exponential nonlinearity, J. Anal. Math. 59 (1992), 251-272 | Zbl

[28] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer-Verlag (2001)

[29] L. Zhang, Blowup solutions of some nonlinear elliptic equations involving exponential nonlinearities, Comm. Math. Phys. 268 no. 1 (2006), 105-133 | Zbl

[30] L. Zhang, Asymptotic behavior of blowup solutions for elliptic equations with exponential nonlinearity and singular data, Commun. Contemp. Math. 11 no. 3 (2009), 395-411 | Zbl