On the classification of solutions of the Lane-Emden equation on unbounded domains of RN

Journal des Mathematiques Pures et Appliquees

Volumn 87, Issue 5, 2007, Pages 537-561

  Farina, Alberto ^a  

^a UNIVERSITÉ DE PICARDIE JULES VERNE   (France)

Author keywords

Classification and qualitative properties of solutions; Critical exponents; Stability

Indexed keywords

EID: 34248165960     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matpur.2007.03.001     Document Type: Article

References (42)

1. 3 and a conjecture of De Giorgi. J. Amer. Math. Soc. 13 4 (2000) 725-739
2. 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
3. Bahri A., and Lions P.-L. Solutions of superlinear elliptic equations and their Morse indices. Comm. Pure Appl. Math. 45 9 (1992) 1205-1215
4. Berestycki H., Caffarelli L., and Nirenberg L. Further qualitative properties for elliptic equations in unbounded domains. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 1-2 (1997) 69-94 (1998)
5. Berestycki H., Capuzzo-Dolcetta I., and Nirenberg L. Superlinear indefinite elliptic problems and nonlinear Liouville theorems. Topol. Methods Nonlinear Anal. 4 1 (1994) 59-78
6. Bidaut-Véron M.-F., and Véron L. Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations. Invent. Math. 106 3 (1991) 489-539
7. Bidaut-Véron M.-F., and Véron L. Erratum: "Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations" [Invent. Math. 106 (3) (1991) 489-539; MR1134481 (93a:35045)]. Invent. Math. 112 2 (1993) 445
8. Brézis H. Elliptic equations with limiting Sobolev exponents-the impact of topology. Frontiers of the Mathematical Sciences: 1985 (New York, 1985). Comm. Pure Appl. Math. 39 suppl. (1986) S17-S39
9. N. C. R. Math. Acad. Sci. Paris 338 10 (2004) 769-774
10. Caffarelli L.A., Gidas B., and Spruck J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42 3 (1989) 271-297
11. Chandrasekhar S. An Introduction to the Study of Stellar Structure (1957), Dover Publications, Inc., New York
12. 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
13. Coron J.-M. Topologie et cas limite des injections de Sobolev (Topology and limit case of Sobolev embeddings). C. R. Acad. Sci. Paris Sér. I, Math. 299 7 (1984) 209-212
14. Cwikel M. Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. (2) 106 1 (1977) 93-100
15. Dancer E.N. Some notes on the method of moving planes. Bull. Austral. Math. Soc. 46 3 (1992) 425-434
16. Dancer E.N. Superlinear problems on domains with holes of asymptotic shape and exterior problems. Math. Z. 229 3 (1998) 475-491
17. N and the primary branch of some non-self-adjoint convex problems. Differential Integral Equations 17 9-10 (2004) 961-970
18. N. Comm. Math. Phys. 107 2 (1986) 331-335
19. Esteban M.J., and Lions P.-L. Existence and nonexistence results for semilinear elliptic problems in unbounded domains. Proc. Roy. Soc. Edinburgh Sect. A 93 1-2 (1982/83) 1-14
20. N. C. R. Math. Acad. Sci. Paris 341 7 (2005) 415-418
21. Fischer-Colbrie D., and Schoen R. The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math. 33 2 (1980) 199-211
22. Fowler R.H. Further studies of Emden's and similar differential equations. Q. J. Math., Oxf. Ser. 2 (1931) 259-288
23. Gidas B., Ni W.M., and Nirenberg L. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 3 (1979) 209-243
24. Gidas B., and Spruck J. A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations 6 8 (1981) 883-901
25. Gidas B., and Spruck J. Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34 4 (1981) 525-598
26. Gilbarg D., and Trudinger N.S. Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Classics in Mathematics (2001), Springer-Verlag, Berlin
27. Joseph D.D., and Lundgren T.S. Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal. 49 (1972/73) 241-269
28. Kavian O. Introduction à la théorie des points critiques et applications aux problèmes elliptiques (Introduction to critical point theory and applications to elliptic problems). Mathématiques et Applications (Mathematics and Applications) vol. 13 (1993), Springer-Verlag, Paris, Berlin (in French)
29. Kazdan J., and Warner F.W. Remarks on some quasilinear elliptic equations. Comm. Pure Appl. Math. 28 5 (1975) 567-597
30. Li P., and Yau S.-T. On the Schrödinger equation and the eigenvalue problem. Comm. Math. Phys. 88 3 (1983) 309-318
31. Lieb E. Bounds on the eigenvalues of the Laplace and Schrödinger operators. Bull. Amer. Math. Soc. 82 5 (1976) 751-753
32. Molle R., and Passaseo D. Multiple solutions of supercritical elliptic problems in perturbed domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 3 (2006) 389-405
33. Moss W.F., and Piepenbrink J. Positive solutions of elliptic equations. Pacific J. Math. 75 1 (1978) 219-226
34. Obata M. The conjectures on conformal transformations of Riemannian manifolds. J. Differential Geometry 6 (1971/72) 247-258
35. Passaseo D. Nontrivial solutions of elliptic equations with supercritical exponent in contractible domains. Duke Math. J. 92 2 (1998) 429-457
36. Pohozaev S.I. On the eigenfunctions of the equation Δ u + λ f (u) = 0. Dokl. Akad. Nauk SSSR 165 (1965) 36-39 English translation in
37. Pohozaev S.I. On the eigenfunctions of the equation Δ u + λ f (u) = 0. Soviet Math. Dokl. 6 (1965) 1408-1411
38. Rozenbljum G.V. Distribution of the discrete spectrum of singular differential operators. Dokl. Akad. Nauk SSSR 202 (1972) 1012-1015 English translation
39. Rozenbljum G.V. Distribution of the discrete spectrum of singular differential operators. Soviet Math. Dokl. 13 (1972) 245-249
40. Serrin J. Local behavior of solutions of quasi-linear equations. Acta Math. 111 (1964) 247-302
41. Struwe M. Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (1990), Springer-Verlag, Berlin/Heidelberg
42. Wang X. On the Cauchy problem for reaction-diffusion equations. Trans. Amer. Math. Soc. 337 2 (1993) 549-590