On the existence of a positive solution of semilinear elliptic equations in unbounded domains
Annales de l'I.H.P. Analyse non linéaire, Volume 14 (1997) no. 3, pp. 365-413.
@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},
pages = {365--413},
publisher = {Gauthier-Villars},
volume = {14},
number = {3},
year = {1997},
zbl = {0883.35045},
mrnumber = {1450954},
language = {en},
url = {http://www.numdam.org/item/AIHPC_1997__14_3_365_0/}
}
TY  - JOUR
AU  - Bahri, Abbas
AU  - Lions, Pierre-Louis
TI  - On the existence of a positive solution of semilinear elliptic equations in unbounded domains
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 1997
DA  - 1997///
SP  - 365
EP  - 413
VL  - 14
IS  - 3
PB  - Gauthier-Villars
UR  - http://www.numdam.org/item/AIHPC_1997__14_3_365_0/
UR  - https://zbmath.org/?q=an%3A0883.35045
UR  - https://www.ams.org/mathscinet-getitem?mr=1450954
LA  - en
ID  - AIHPC_1997__14_3_365_0
ER  - 
%0 Journal Article
%A Bahri, Abbas
%A Lions, Pierre-Louis
%T On the existence of a positive solution of semilinear elliptic equations in unbounded domains
%J Annales de l'I.H.P. Analyse non linéaire
%D 1997
%P 365-413
%V 14
%N 3
%I Gauthier-Villars
%G en
%F 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, Volume 14 (1997) no. 3, pp. 365-413. http://www.numdam.org/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

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

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

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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

[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

[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 | Zbl

[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

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

[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 | Zbl

[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

[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 | Zbl

[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

[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 | Zbl

[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

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

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

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

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

[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 | Zbl

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 | Zbl

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

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

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

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

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

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

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

[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

[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 | Zbl