Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 24, Issue 2, 2007, Pages 325-340

  Pistoia, Angela ^a   Weth, Tobias ^b  

^a SAPIENZA UNIVERSITY OF ROME   (Italy)

^b JUSTUS LIEBIG UNIVERSITY   (Germany)

Author keywords

Critical Sobolev exponent; Multiple blow up; Sign changing solution

Indexed keywords

COMPUTATIONAL GEOMETRY; LINEAR EQUATIONS; PARAMETER ESTIMATION;

CRITICAL SOBOLEV EXPONENT; MULTIPLE BLOW UP; SIGN CHANGING SOLUTION;

PROBLEM SOLVING;

EID: 33947159050     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.anihpc.2006.03.002     Document Type: Article

References (43)

1. Atkinson F.V., Brezis H., and Peletier L.A. Solutions d'équations elliptiques avec exposant de Sobolev critique qui changent de signe. C. R. Acad. Sci. Paris 306 (1988) 711-714
2. Atkinson F.V., Brezis H., and Peletier L.A. Nodal solutions of elliptic equations with critical Sobolev exponents. J. Differential Equations 85 (1990) 151-170
3. Aubin T. Problèmes isoperimetriques et espaces de Sobolev. J. Differential Geom. 11 (1976) 573-598
4. Bahri A. Critical Points at Infinity in Some Variational Problems. Pitman Research Notes in Mathematics Series vol. 182 (1989), Longman
5. Bahri A., and 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
6. Bahri A., Li 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 (1995) 67-93
7. Bartsch T. Critical point theory on partially ordered Hilbert spaces. J. Funct. Anal. 186 (2001) 117-152
8. Bartsch T., and Wang Z.-Q. On the existence of sign changing solutions for semilinear Dirichlet problems. Topol. Methods Nonlinear Anal. 7 (1996) 115-131
9. Bartsch T., and Weth T. A note on additional properties of sign changing solutions to superlinear elliptic equations. Topol. Methods Nonlinear Anal. 22 (2003) 1-14
10. Bartsch T., Micheletti A.M., and Pistoia A. On the existence and the profile of nodal solutions of elliptic equations involving critical growth. Calc. Var. Partial Differential Equations 26 3 (2006) 265-282
11. Ben Ayed M., El Mehdi K., Grossi M., and Rey O. A nonexistence result of single peaked solutions to a supercritical nonlinear problem. Comm. Contemp. Math. 5 2 (2003) 179-195
12. Bianchi G., and Egnell H. A note on the Sobolev inequality. J. Funct. Anal. 100 (1991) 18-24
13. Brézis H., and Peletier L.A. Asymptotics for elliptic equations involving critical growth. Partial Differential Equations and the Calculus of Variations, vol. I. Progr. Nonlinear Differential Equations Appl. vol. 1 (1996), Birkhäuser, Boston, MA 149-192
14. Caffarelli L., Gidas B., and Spruck J. Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42 (1989) 271-297
15. Castro A., and Clapp M. The effect of the domain topology on the number of minimal nodal solutions of an elliptic equation at critical growth in a symmetric domain. Nonlinearity 16 (2003) 579-590
16. Castro A., Cossio J., and Neuberger J.M. A sign-changing solution for a superlinear Dirichlet problem. Rocky Mountain J. Math. 27 (1997) 1041-1053
17. Cerami G., Solimini S., and Struwe M. Some existence results for superlinear elliptic boundary value problems involving critical exponents. J. Funct. Anal. 69 (1986) 298-306
18. Clapp M., and Weth T. Multiple solutions for the Brezis-Nirenberg problem. Adv. Differential Equations 10 4 (2005) 463-480
19. Clapp M., and Weth T. Minimal nodal solutions of the pure critical exponent problem on a symmetric domain. Calc. Var. Partial Differential Equations 21 1 (2004) 1-14
20. del Pino M., Dolbeault J., and Musso M. "Bubble-tower" radial solutions in the slightly supercritical Brezis-Nirenberg problem. J. Differential Equations 193 2 (2003) 280-306
21. del Pino M., Dolbeault J., and Musso M. The Brezis-Nirenberg problem near criticality in dimension 3. J. Math. Pures Appl. (9) 83 12 (2004) 1405-1456
22. del Pino M., Felmer P., and Musso M. Two-bubble solutions in the super-critical Bahri-Coron's problem. Calc. Var. Partial Differential Equations 16 (2003) 113-145
23. 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 (2003) 513-521
24. del Pino M., Musso M., and Pistoia A. Super-critical boundary bubbling in a semilinear Neumann problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 1 (2005) 45-82
25. N. Comm. Math. Phys. 107 (1986) 331-335
26. Flucher M., and Wei J. Semilinear Dirichlet problem with nearly critical exponent, asymptotic location of hot spots. Manuscripta Math. 94 (1997) 337-346
27. Fortunato D., and Jannelli E. Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains. Proc. Roy. Soc. Edinburgh 105 (1987) 205-213
28. Ge Y., Jing R., and Pacard F. Bubble towers for supercritical semilinear elliptic equations. J. Funct. Anal. 221 2 (2005) 251-302
29. Han Z.C. Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 159-174
30. Hebey E., and Vaugon M. Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev growth. J. Funct. Anal. 119 (1994) 298-318
31. N. Comm. Pure Appl. Anal. 4 1 (2005) 143-164
32. Kazdan J., and Warner F.W. Remarks on some quasilinear elliptic equations. Comm. Pure Appl. Math. 28 (1975) 567-597
33. n and related problems. I. J. Differential Equations 120 2 (1995) 319-410
34. Micheletti A.M., and Pistoia A. On the effect of the domain geometry on the existence of sign changing solutions to elliptic problems with critical and supercritical growth. Nonlinearity 17 3 (2004) 851-866
35. Molle R., and Passaseo D. Positive solutions for slightly super-critical elliptic equations in contractible domains. C. R. Math. Acad. Sci. Paris, Ser. I 335 (2002) 459-462
36. Müller-Pfeiffer E. On the number of nodal domains for elliptic differential operators. J. London Math. Soc. (2) 31 (1985) 91-100
37. Musso M., and Pistoia A. Multispike solutions for a nonlinear elliptic problem involving critical Sobolev exponent. Indiana Univ. Math. J. 5 (2002) 541-579
38. A. Pistoia, O. Rey, Multiplicity of solutions to the supercritical Bahri-Coron's problem in pierced domains, Adv. Differential Equations, in press
39. Pohožaev S.I. On the eigenfunctions of the equation Δ u + λ f (u) = 0. Dokl. Akad. Nauk SSSR 165 (1965) 36-39 (in Russian)
40. Rey O. Blow-up points of solutions to elliptic equations with limiting nonlinearity. Differential Integral Equations 4 (1991) 1155-1167
41. 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
42. Rey O. Proof of two conjectures of H. Brezis and L.A. Peletier. Manuscripta Math. 65 (1989) 19-37
43. Talenti G. Best constants in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976) 353-372