Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent

Journal of Differential Equations

Volumn 227, Issue 1, 2006, Pages 29-68

  Esposito, Pierpaolo ^a   Musso, Monica ^b,c   Pistoia, Angela ^d  

^a ROMA TRE UNIVERSITY   (Italy)

^b PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE   (Chile)

^c POLITECNICO DI TORINO   (Italy)

^d SAPIENZA UNIVERSITY OF ROME   (Italy)

Author keywords

Concentrating solutions; Finite dimensional reduction; Green's function; Large exponent

Indexed keywords

EID: 33747419612     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2006.01.023     Document Type: Article

References (29)

1. Adimurthi, and Grossi M. Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity. Proc. Amer. Math. Soc. 132 4 (2004) 1013-1019
2. Bahri A., Li Y.Y., and Rey O. On a variational problem with lack of compactness: The topological effect of the critical points at infinity. Calc. Var. Partial Differential Equations 3 1 (1995) 67-93
3. Baraket S., and Pacard F. Construction of singular limits for a semilinear elliptic equation in dimension 2. Calc. Var. Partial Differential Equations 6 1 (1998) 1-38
4. Byeon J. Nonlinear elliptic problems on singularly perturbed domains. Proc. Roy. Soc. Edinburgh Sect. A 131 5 (2001) 1023-1037
5. Chae D., and Imanuvilov O. The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory. Comm. Math. Phys. 215 (2000) 119-142
6. Chen W., and Li C. Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63 (1991) 615-623
7. Dancer E.N. The effect of domain shape on the number of positive solutions of certain nonlinear equations. J. Differential Equations 74 1 (1988) 120-156
8. Dancer E.N., and Yan S. A singularly perturbed elliptic problem in bounded domains with nontrivial topology. Adv. Differential Equations 4 3 (1999) 347-368
9. del Pino M., Felmer P., and Musso M. Multi-peak solutions for super-critical elliptic problems in domains with small holes. J. Differential Equations 182 2 (2002) 511-540
10. del Pino M., Felmer P., and Musso M. Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries. Bull. London Math. Soc. 35 4 (2003) 513-521
11. del Pino M., Felmer P., and Wei J. On the role of distance function in some singular perturbation problems. Comm. Partial Differential Equations 25 1/2 (2000) 155-177
12. del Pino M., Kowalczyk M., and Musso M. Singular limits in Liouville-type equations. Calc. Var. Partial Differential Equations 24 1 (2005) 47-81
13. Esposito P. Blow up solutions for a Liouville equation with singular data. In: Chen C.-C., Chipot M., and Lin C.-S. (Eds). Recent Advances in Elliptic and Parabolic Problems. Taiwan, 2004 (2005), World Sientific, Hackensack, NJ 39-57
14. Esposito P., Grossi M., and Pistoia A. On the existence of blowing-up solutions for a mean field equation. Ann. Inst. H. Poincaré Analyse Non Linéaire 22 2 (2005) 227-257
15. Esposito P., and Mancini G. A prescribed scalar curvature-type equation: Almost critical manifolds and multiple solutions. J. Funct. Anal. 205 2 (2003) 306-356
16. El Mehdi K., and Grossi M. Asymptotic estimates and qualitative properties of an elliptic problem in dimension two. Adv. Nonlinear Stud. 4 1 (2004) 15-36
17. Flucher M., and Wei J. Semilinear Dirichlet problem with nearly critical exponent, asymptotic location of hot spots. Manuscripta Math. 94 3 (1997) 337-346
18. Grossi M., and Pistoia A. On the effect of critical points of distance function in super-linear elliptic problems. Adv. Differential Equations 5 10-12 (2000) 1397-1420
19. Han Z.C. Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincaré Analyse Non Linéaire 8 (1991) 159-174
20. Li Y.Y., and Nirenberg L. The Dirichlet problem for singularly perturbed elliptic equations. Comm. Pure Appl. Math. 51 11/12 (1998) 1445-1490
21. Musso M., and Pistoia A. Multispike solutions for a nonlinear elliptic problem involving the critical Sobolev exponent. Indiana Univ. Math. J. 51 3 (2002) 541-579
22. Ni W.N., and Wei J. On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. Comm. Pure Appl. Math. 48 7 (1995) 731-768
23. - 1 = 0 in noncontractible domains. Topol. Methods Nonlinear Anal. 2 (1993) 343-366
24. A. Pistoia, O. Rey, Multiplicity of solutions to the supercritical Bahri-Coron's problem in pierced domains, Adv. Differential Equations, submitted for publication
25. Rey O. The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent. J. Funct. Anal. 89 (1990) 1-52
26. Rey O. A multiplicity result for a variational problem with lack of compactness. Nonlinear Anal. 13 (1989) 1241-1249
27. Ren X., and Wei J. On a two-dimensional elliptic problem with large exponent in nonlinearity. Trans. Amer. Math. Soc. 343 2 (1994) 749-763
28. Ren X., and Wei J. Single point condensation and least energy solutions. Proc. Amer. Math. Soc. 124 (1996) 111-120
29. Wei J. On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem. J. Differential Equations 129 2 (1996) 315-333