Abstract
We formulate the concentration-compactness principle at infinity for both subcritical and critical case. We show some applications to the existence theory of semilinear elliptic equations involving critical and subcritical Sobolev exponents.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. P. Garcia Azorero, P. Alonso: Multiplicity of solutions for elliptic problems with critical exponent or with non symmetric term. Trans. Am. Math. Soc.323 (2) (1991) 887–897
J. P. Garcia Azorero, P. Alonso: On limits of solutions of elliptic problems with nearly critical exponent. Commun in PDE17 (ii & 12) (1992) 2113–2126
A. Bahri, Yan Yan Li: On a min-max procedure for the existence of a positive solution for certain scalar field equations in ℝ N . Revista Mat. Iberoamericana6 (1, 2) (1990) 1–15
A.K. Ben-Naouma, C. Troestler, M. Willem: Extrema problems with critical Sobolev exponents on unbounded domains. Institut de Mathématique Pure et Appliquée. Université Catholique de Louvain (preprint)
V. Benci, G. Cerami: Positive solutions of some nonlinear elliptic problems in exterior domains. Arch. Rat. Mech. Anal.99 (1987) 283–300
V. Benci, G. Cerami: Existence of positive solutions of the equation −Δu+a(x)u=u N+2/N−2 in ℝ N . J. Funct Anal.88 (1990) 90–117
G. Bianchi, J. Chabrowski, A. Szulkin: Symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent. Nonlinear Analysis, TMA (to appear)
H. Brezis, L. Nirenberg: Positive solutions od nonliner elliptic equations involving critical Sobolev exponent. Commun. Pure Appl. Math.36 (1983) 437–477
Cao Daomin: Yamabe problem in ℝ n and related problems. Acta Mat. Scientia10(2) (1990) 201–216
Dao-Min Cao: Multiple solutions of a semilinear elliptic equation in ℝN. Ann. Inst. H. Poincaré, Analyse non linéaire10 (6) (1993) 593–604
J. Chabrowski: On entire solutions of the p-Laplacian. Proc. Centre for Maths and its Applications, Australian National University26 (1991) 27–61
I. Ekeland: Nonconvex minimization problems. Bull. Am. Math. Soc.1 (1979) 443–474
M.J. Esteban: Nonlinear problems in strip-like domains: symmetry of positive vortex rings. Nonlinear Analysis TMA7 (4) (1983) 365–379
M.K. Kwong: Uniqueness of positive solution ofΔu −u+u p=0. Arch. Rat. Mech. Anal.105 (1989) 243–266
P.L. Lions: The concentration-compactness principle in the calculus of variations, the limit case, Part 1,2. Revista Mat. Iberoamericana1 (2, 3) (1985) 145–201, 45–121
P.L. Lions: The concentration-compactness principle in the calculus of variations, locally compact case, Part 1,2. Ann. Inst. H. Poincaré1 (1&4) (1984) 109–145, 223–283
M. Struwe: Variational Methods. Springer, Berlin Heidelberg 1990
C. Stuart: Bifurcation inL p(ℝN) for a semilinear elliptic equation. Proc. London Math. Soc. (3)57 (1988) 511–541
G. Talenti: Best constant in Sobolev inequality. Ann. Math. Pura Appl.101 (1976) 353–372
Weng-Ching Lien, Shyuh-Yaur Tzeng, Hwai-Chiuan Wang: Existence of solutions of semilinear elliptic problems on unbounded domains. Differ. Int. Equations6 (6) (1993) 128–140
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chabrowski, J. Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents. Calc. Var 3, 493–512 (1995). https://doi.org/10.1007/BF01187898
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01187898