On the existence of a positive solution of semilinear elliptic equations in unbounded domains
Bahri, Abbas ; Lions, Pierre-Louis
Annales de l'I.H.P. Analyse non linéaire, Tome 14 (1997), p. 365-413 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU) 
@article{AIHPC_1997__14_3_365_0,
     author = {Bahri, Abbas and Lions, Pierre-Louis},
     title = {On the existence of a positive solution of semilinear elliptic equations in unbounded domains},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {14},
     year = {1997},
     pages = {365-413},
     mrnumber = {1450954},
     zbl = {0883.35045},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_1997__14_3_365_0}
}

Bahri, Abbas; Lions, Pierre-Louis. On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Annales de l'I.H.P. Analyse non linéaire, Tome 14 (1997) pp. 365-413. http://gdmltest.u-ga.fr/item/AIHPC_1997__14_3_365_0/

[1] A. Bahri, Pseudo-Orbits of contact forms, Pitman Research Notes in Mathematics, Longman London, Vol. 173. | Zbl 0677.58002

[2] A. Bahri, Critical points at infinity in some variational problems, Pitman Research Notes in mathematics, Longman, London, Vol. 182. | Zbl 0676.58021

[3] A. Bahri, Topological results on a certain class of functionals and applications, J. Funct. Anal., Vol. 41, 1981, pp. 397-427. | MR 619960 | Zbl 0499.35050

[4] A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc., Vol. 297, 1987, pp. 1-32. | MR 621969 | Zbl 0476.35030

[5] A. Bahri and J.M. Coron, On a non-linear ellepttic equation involving the critical Sobolev exponent; the effect of the topology of the domain, Comm. Pure and Appl. Math., Vol. 41, 1988, pp. 253-294. | MR 929280 | Zbl 0649.35033

[6] V. Benci and G. Cerami, Positive solutions of semilinear elliptic problems in exterior domains. Preprint.

[7] H. Berestycki and P.-L. Lions, Nonlinear scalar fields equations, Arch. Rat. Mech. Anal., Vol. I 82, 1983, pp. 313-346; Vol. II 82, 1983, pp. 347-376. | Zbl 0533.35029

[8] M. Berger, On the existence and structure of stationary states for a nonlinear Klein-Gordon equation, J. Funct. Anal., Vol. 9, 1972, pp. 249-261. | MR 299966 | Zbl 0224.35061

[9] G. Bredon, Introduction to Compact transformation Groups, New York-Academic Press, 1972. | MR 413144 | Zbl 0246.57017

[10] H. Brezis and J.M. Coron, convergence of solutions of H-systems or how to blow bubbles, Arch. Rational Mech. Anal., Vol. 89, 1985, pp. 21-56. | MR 784102 | Zbl 0584.49024

[11] C.V. Coffman, Uniqueness of the groundstate solution for Δu - u + u3 = 0 and a variational characterization of other solutions, Arch. Rat. Mech. Anal., Vol. 46, 1982, pp. 81-95. | MR 333489 | Zbl 0249.35029

[12] C.V. Coffman and M. Marcus, personal communication.

[13] C.V. Coffman and M. Marcus, Existence theorems for superlinear elliptic Dirichlet problems in exterior domains, Nonlinear Analysis and Its Applications, Part 2, Vol. 45; AMS, Providence, 1983. | Zbl 0596.35048

[14] S. Coleman, V. Glaser and A. Mawhin, Action minima among solutions to a class of Euclidian scalar field equations, Comm. Math. Phys., Vol. 58, 1978, pp. 211-221. | MR 468913

[15] J.M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris, I, Vol. 299, 1984, pp. 209-212. | MR 762722 | Zbl 0569.35032

[16] W.Y. Ding and W.M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rat. Mech Anal. | Zbl 0616.35029

[17] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc., Vol. I, 1979, pp. 443-479. | MR 526967 | Zbl 0441.49011

[18] M.J. Esteban and P.-L. Lions, Existence and nonexistence results for semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edim, Vol. 93, 1982, pp. 1-14; C. R. Acad. Sci. Paris, Vol. 290, 1980, pp. 1083-1085. | MR 688279 | Zbl 0506.35035

[19] M.J. Esteban and P.-L. Lions, Γ-convergence and the concentration-compactness method for some variational problems with lack of compactness. Preprint. | MR 977055

[20] B. Gidas, W.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., Vol. 68, 1979, pp. 209-243. | MR 544879 | Zbl 0425.35020

[21] B. Gidas, W.M. Ni And L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn, Advances in Math. Supplementary Studies, Vol. 7, 1981, pp. 369-402. | Zbl 0469.35052

[22] M.K. Kwong, Uniqueness positive solutions of Δu - u + up = 0 in Rn, Arch. Rat. Mech. Anal., Vol. 105, 1985, pp. 243-266. | MR 969899 | Zbl 0676.35032

[23] P.-L. Lions, The concentration-compactness principle in the Calculus of Variations, The locally compact case, Ann. Inst. H. Poincaré, Vol. I 1, 1984, pp. 109-145; Vol. II 1, 1984, pp. 223-283, see also, C. R. Acad. Sci. Paris 294, Vol. 1, pp. 223-283; 1982, pp. 261-264; Contributions to nonlinear partial differential equations, Pitman, London 1983. | Numdam | Zbl 0541.49009

[24] P.-L. Lions, On positive solutions of semilinear elliptic equations in unbounded domains. Preprint. | MR 956083

[25] P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys. | MR 879032 | Zbl 0618.35111

[26] P.-L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal., Vol. 49, 1982, pp. 315-334. | MR 683027 | Zbl 0501.46032

[27] P.-L. Lions, Minimization problems in L1, J. Funct. Anal., Vol. 49, 1982, pp. 315-334. | Zbl 0501.46032

[28] P.-L. Lions, The concentration-compactness principle in the Calculus of Variations. The limit case, Riv. Mat. Ibereamericana, Vol. I 1, 1985, pp. 145-201; Vol. II 1, 1985, | MR 834360 | Zbl 0704.49005

pp. 45-121; see also Seminaire, Goulaouic-Meyer-Schwartz, 1982-1983, exposé XIV, École Polytechnique, Palaiseau and C. R. Acad. Sci. Paris; 296, 1983, pp. 645-648. | Numdam

[29] K. Mac Leod and J. Serrin, Uniqueness of solutions of semilinear Poisson equations, Proc. Nat. Acad. Sci. USA, Vol. 78, 1981, pp. 6592-6595. | MR 634934 | Zbl 0474.35047

[30] Milnor, Morse theory, Annals of Mathematical Studies, Princeton Univ. Press, Study No. 52. | Zbl 0108.10401

[31] Z. Nehari, On a nonlinear differential equation arising in nuclear physics, Proc. Roy. Irish Acad., Vol. 62, 1963, pp. 117-135. | MR 165176 | Zbl 0124.30204

[32] P.H. Rabinowitz, Variational methods for nonlinear eigenvalue problems. In Eigenvalues of Nonlinear Problems, Edis. Cremonese, Rome, 1972. | MR 464299

[33] G.H. Ryder, Boundary value problems for a class of nonlinear differential equations, Proc. J. Math., Vol. 22, 1967, pp. 477-503. | MR 219794 | Zbl 0152.28303

[34] P. Saks and K. Uhlenbeck, The existence of minimal immersions of 2-sphere, Ann. Math., Vol. 113, 1981, pp. 1-24. | MR 604040 | Zbl 0462.58014

[35] W. Strauss, Existence of solitary vaves in higher dimensions, Comm. Math. Phys., Vol. 55, 1977, pp. 149-162. | MR 454365 | Zbl 0356.35028

[36] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., Vol. 187, 1984, pp. 511-517. | MR 760051 | Zbl 0535.35025

[37] C.H. Taubes, The existence of a non-minimal solution to the SU(2) Yang-Mills-Higgs equations on R3 I, Comm. Math. Physics, Vol. 86, 1982, pp. 257-298. | Zbl 0514.58016

[38] C.H. Taubes, The existence of a non-minimal solution to the SU(2) Yang-Mills-Higgs equations on R3 II, Comm. Math. Physics, Vol. 86, 1982, pp. 299-320. | MR 677000 | Zbl 0514.58017