Symmetry and nonexistence of low Morse index solutions in unbounded domains

Journal des Mathematiques Pures et Appliquees

Volumn 93, Issue 5, 2010, Pages 536-558

  Gladiali, Francesca ^a   Pacella, Filomena ^b   Weth, Tobias ^c  

^a UNIVERSITY OF SASSARI   (Italy)

^b SAPIENZA UNIVERSITY OF ROME   (Italy)

^c GOETHE UNIVERSITY   (Germany)

Author keywords

Nonexistence of solutions; Semilinear elliptic equations; Symmetry results; Unbounded domains

Indexed keywords

EID: 77951209137     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matpur.2009.08.003     Document Type: Article

References (29)

1. Aftalion A., Pacella F. Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains. C. R. Math. Acad. Sci. Paris 2004, 339(5):339-344.
2. Bartsch T., Liu Z., Weth T. Sign changing solutions of superlinear Schrödinger equations. Comm. Partial Differential Equations 2004, 29:25-42.
3. Bartsch T., Weth T., Willem M. Partial symmetry of least energy nodal solutions to some variational problems. J. Anal. Math. 2005, 96:1-18.
4. Bartsch T., Willem M. Infinitely many nonradial solutions of a Euclidean scalar field equation. J. Funct. Anal. 1993, 117:447-460.
5. Berestycki H., Caffarelli L., Nirenberg L. Further qualitative properties for elliptic equations in unbounded domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1997, 25(1-2):69-94.
6. Brezis H. Analyse fonctionnelle. Théorie et applications 1983, Masson, Paris.
7. Brock F. Positivity and radial symmetry of solutions to some variational problems in RN. J. Math. Anal. Appl. 2004, 296(1):226-243.
8. Cabré X., Capella A. On the stability of radial solutions of semilinear elliptic equations in all of Rn. C. R. Math. Acad. Sci. Paris 2004, 338(10):769-774.
9. Chen C.C., Lin C.S. Local behavior of singular positive solutions of semilinear elliptic equations with Sobolev exponent. Duke Math. J. 1995, 78(2):315-334.
10. Damascelli L., Gladiali F. Some nonexistence results for positive solutions of elliptic equations in unbounded domains. Rev. Mat. Iberoamericana 2004, 20(1):67-86.
11. Dancer E.N. Stable and finite Morse index solutions on Rn or on bounded domains with small diffusion. Trans. Amer. Math. Soc. 2005, 357(3):1225-1243.
12. Dancer E.N. Stable and finite Morse index solutions on Rn or on bounded domains with small diffusion. II. Indiana Univ. Math. J. 2004, 53(1):97-108.
13. Dancer E.N. Stable and not too unstable solutions on Rn for small diffusion. Fields Inst. Commun. 2006, vol. 48:67-93. Amer. Math. Soc., Providence, RI.
14. Dancer E.N., Farina A. On the classification of solutions of -Δu=eu on RN: Stability outside a compact set. Proc. Amer. Math. Soc. 2009, 137(4):1333-1338.
15. Ding W.Y. On a conformally invariant elliptic equation on RN. Comm. Math. Phys. 1986, 107:331-335.
16. Dupaigne L., Farina A. Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities. Nonlinear Anal. 2009, 70(8):2882-2888.
17. Farina A. On the classification of solutions of the Lane-Emden equation on unbounded domains of RN. J. Math. Pures Appl. (9) 2007, 87(5):537-561.
18. Gidas B., Ni Wei Ming, Nirenberg L. Symmetry of positive solutions of nonlinear elliptic equations in Rn. Adv. in Math. Suppl. Stud. 1981, vol. 7a:369-402. Academic Press, New York, London.
19. Gidas B., Spruck J. Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 1981, 34:525-598.
20. Jones C.K.R.T., Küpper T. On the infinitely many solutions of a semilinear elliptic equation. SIAM J. Math. Anal. 1986, 17:803-835.
21. Kwong M.K. Uniqueness of positive solutions of Δu-u+up=0 in Rn. Arch. Rational Mech. Anal. 1989, 105:243-266.
22. Maris M. On the symmetry of minimizers. Arch. Rational Mech. Anal. 2009, 192:311-330.
23. Montefusco E. Axial symmetry of solutions to semilinear elliptic equations in unbounded domains. Proc. Roy. Soc. Edinburgh Sect. A 2003, 133(5):1175-1192.
24. Pacella F. Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities. J. Funct. Anal. 2002, 192(1):271-282.
25. Pacella F., Weth T. Symmetry of solutions to semilinear elliptic equations via Morse index. Proc. Amer. Math. Soc. 2007, 135(6):1753-1762.
26. Pohožaev S.I. On the eigenfunctions of the equation Δu+λf(u)=0. Dokl. Akad. Nauk SSSR 1965, 165:36-39.
27. Smets D., Willem M. Partial symmetry and asymptotic behavior for some elliptic variational problems. Calc. Var. Partial Differential Equations 2003, 18(1):57-75.
28. Van Schaftingen J., Willem M. Symmetry of solutions of semilinear elliptic problems. J. Eur. Math. Soc. (JEMS) 2008, 10(2):439-456.
29. Weth T. Energy bounds for entire nodal solutions of autonomous superlinear equations. Calc. Var. Partial Differential Equations 2006, 27:421-437.