Singular limits for the bi-laplacian operator with exponential nonlinearity in R 4

Mónica Clapp; Claudio Muñoz; Monica Musso

Annales de l'I.H.P. Analyse non linéaire (2008)

  • Volume: 25, Issue: 5, page 1015-1041
  • ISSN: 0294-1449

How to cite


Clapp, Mónica, Muñoz, Claudio, and Musso, Monica. "Singular limits for the bi-laplacian operator with exponential nonlinearity in ${R}^{4}$." Annales de l'I.H.P. Analyse non linéaire 25.5 (2008): 1015-1041. <>.

author = {Clapp, Mónica, Muñoz, Claudio, Musso, Monica},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {Degenerating exponential nonlinearity; Variational structure; Blow-up},
language = {eng},
number = {5},
pages = {1015-1041},
publisher = {Elsevier},
title = {Singular limits for the bi-laplacian operator with exponential nonlinearity in $\{R\}^\{4\}$},
url = {},
volume = {25},
year = {2008},

AU - Clapp, Mónica
AU - Muñoz, Claudio
AU - Musso, Monica
TI - Singular limits for the bi-laplacian operator with exponential nonlinearity in ${R}^{4}$
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2008
PB - Elsevier
VL - 25
IS - 5
SP - 1015
EP - 1041
LA - eng
KW - Degenerating exponential nonlinearity; Variational structure; Blow-up
UR -
ER -


  1. [1] Adimurthi, Robert F., Struwe M., Concentration phenomena for Liouville's equation in dimension four, J. Eur. Math. Soc. (JEMS)8 (2) (2006) 171-180. Zbl05053356MR2239297
  2. [2] Bahri A., Coron J.M., On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math.41 (3) (1988) 253-294. Zbl0649.35033MR929280
  3. [3] Bahri A., Li Y.-Y., Rey O., On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Calc. Var.3 (1) (1995) 67-93. Zbl0814.35032MR1384837
  4. [4] Baraket S., Dammak M., Pacard F., Ouni T., Singular limits for 4-dimensional semilinear elliptic problems with exponential nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire24 (2007) 875-895. Zbl1132.35038MR2371110
  5. [5] Baraket S., Pacard F., Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. Partial Differential Equations6 (1) (1998) 1-38. Zbl0890.35047
  6. [6] Bartolucci D., Tarantello G., The Liouville equation with singular data: a concentration-compactness principle via a local representation formula, J. Differential Equations185 (1) (2002) 161-180. Zbl1247.35032
  7. [7] Branson T.P., Differential operators canonically associated to a conformal structure, Math. Scand.57 (1985) 293-345. Zbl0596.53009
  8. [8] Brezis H., Merle F., Uniform estimates and blow-up behavior for solutions of - Δ u = V x e u in two dimensions, Comm. Partial Differential Equations16 (8–9) (1991) 1223-1253. Zbl0746.35006
  9. [9] S.Y.A. Chang, On a fourth order operator – the Paneitz operator – in conformal geometry, in: Proceedings of the Conference for the 70th of A.P. Calderon, in press. 
  10. [10] Chang S.Y.A., Yang P.C., On a fourth order curvature invariant, in: Branson T. (Ed.), Spectral Problems in Geometry and Arithmetic, Contemporary Math., vol. 237, Amer. Math. Soc., 1999, pp. 9-28. Zbl0982.53035MR1710786
  11. [11] Chen C.-C., Lin C.-S., Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math.56 (12) (2003) 1667-1727. Zbl1032.58010MR2001443
  12. [12] del Pino M., Felmer P., Musso M., Two-bubble solutions in the super-critical Bahri–Coron's problem, Calc. Var. Partial Differential Equations16 (2) (2003) 113-145. Zbl1142.35421MR1956850
  13. [13] del Pino M., Dolbeault J., Musso M., “Bubble-tower” radial solutions in the slightly supercritical Brezis–Nirenberg problem, J. Differential Equations193 (2) (2003) 280-306. Zbl1140.35413MR1998635
  14. [14] del Pino M., Kowalczyk M., Musso M., Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations24 (2005) 47-81. Zbl1088.35067MR2157850
  15. [15] del Pino M., Kowalczyk M., Musso M., Variational reduction for Ginzburg–Landau vortices, J. Funct. Anal.239 (2) (2006) 497-541. Zbl05083435MR2261336
  16. [16] del Pino M., Muñoz C., The two-dimensional Lazer–McKenna conjecture for an exponential nonlinearity, J. Differential Equations231 (2006) 108-134. Zbl1159.35372MR2287880
  17. [17] Dold A., Lectures on Algebraic Topology, Springer-Verlag, Berlin, 1972. Zbl0234.55001MR415602
  18. [18] Druet O., Robert F., Bubbling phenomena for fourth-order four-dimensional PDEs with exponential growth, Proc. Amer. Math. Soc.134 (3) (2006) 897-908. Zbl1083.58018MR2180908
  19. [19] Z. Djadli, A. Malchiodi, Existence of conformal metrics with constant Q-curvature, Ann. Math., in press. Zbl1186.53050
  20. [20] P. Esposito, A class of Liouville-type equations arising in Chern–Simons vortex theory: asymptotics and construction of blowing-up solutions, Ph.D. Thesis, Universitá di Roma “Tor Vergata”, 2004. 
  21. [21] Esposito P., Grossi M., Pistoia A., On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire22 (2) (2005) 227-257. Zbl1129.35376MR2124164
  22. [22] Esposito P., Musso M., Pistoia A., Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent, J. Differential Equations227 (1) (2006) 29-68. Zbl1254.35083MR2233953
  23. [23] Kazdan J., Warner F., Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. Math.101 (1975) 317-331. Zbl0297.53020MR375153
  24. [24] Li Y.-Y., Shafrir I., Blow-up analysis for solutions of - Δ u = V e u in dimension two, Indiana Univ. Math. J.43 (4) (1994) 1255-1270. Zbl0842.35011MR1322618
  25. [25] Liouville J., Sur l’equation aux difference partielles d 2 log λ d u d v ± λ 2 a 2 = 0 , C. R. Acad. Sci. Paris36 (1853) 71-72. 
  26. [26] Lin C.S., Wei J., Locating the peaks of solutions via the maximum principle. II. A local version of the method of moving planes, Comm. Pure Appl. Math.56 (6) (2003) 784-809. Zbl1121.35310MR1959740
  27. [27] C.S. Lin, J. Wei, Sharp estimates for bubbling solutions of a fourth order mean field equation, Preprint, 2007. Zbl1185.35067MR2394412
  28. [28] C.S. Lin, L.P. Wang, J. Wei, Topological degree for solutions of a fourth order mean field equation, Preprint, 2007. 
  29. [29] Ma L., Wei J., Convergence for a Liouville equation, Comment. Math. Helv.76 (3) (2001) 506-514. Zbl0987.35056MR1854696
  30. [30] Malchiodi A., Struwe M., Q-curvature flow on S 4 , J. Differential Geom.73 (1) (2006) 1-44. Zbl1099.53034MR2217518
  31. [31] Nagasaki K., Suzuki T., Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptotic Anal.3 (2) (1990) 173-188. Zbl0726.35011MR1061665
  32. [32] Paneitz S., Essential unitarization of symplectics and applications to field quantization, J. Funct. Anal.48 (1982) 310-359. Zbl0499.47025MR678176
  33. [33] Pistoia A., Rey O., Multiplicity of solutions to the supercritical Bahri–Coron's problem in pierced domains, Adv. Differential Equations11 (6) (2006) 647-666. Zbl1166.35333MR2238023
  34. [34] Rey O., The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal.89 (1) (1990) 1-52. Zbl0786.35059MR1040954
  35. [35] Suzuki T., Two-dimensional Emden–Fowler equation with exponential nonlinearity, in: Nonlinear Diffusion Equations and their Equilibrium States, 3, Gregynog, 1989, Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992. Zbl0792.35061MR1167859
  36. [36] Tarantello G., A quantization property for blow-up solutions of singular Liouville-type equations, J. Funct. Anal.219 (2) (2005) 368-399. Zbl1174.35379MR2109257
  37. [37] Tarantello G., Analytical aspects of Liouville-type equations with singular sources, in: Stationary Partial Differential Equations, vol. I, Handbook of Differential Equations, North-Holland, Amsterdam, 2004, pp. 491-592. Zbl1129.35408MR2103693
  38. [38] Weston V.H., On the asymptotic solution of a partial differential equation with an exponential nonlinearity, SIAM J. Math. Anal.9 (6) (1978) 1030-1053. Zbl0402.35038MR512508

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