Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-02T20:15:26.771Z Has data issue: false hasContentIssue false
  • English
  • Français

Semilinear elliptic problems with singular potentials in ℝN

Published online by Cambridge University Press:  14 November 2011

Marino Badiale  and
Nicoletta A. Tchou
Marino Badiale
Affiliation:
Universita degli studi di Padova, Dipartimento di Matematica Pura e Applicata, via Belzoni 7, 35131 Padova, Italy
Nicoletta A. Tchou
Affiliation:
Universita degli studi dell'Aquila, Dipartimento di Matematica- loc. Coppito, 67100 L'Aquila, Italy
Get access Rights & Permissions [Opens in a new window]

Synopsis

We study a semilinear elliptic problem on ℝN where a potential is not a bounded function but can also be an infinite measure. We analyse the lack of compactness of the problem, obtaining a structure theorem for Palais-Smale sequences. This result allows us to obtain different kinds of existence theorems by a variational approach.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 4 , 1993 , pp. 593 - 619
Copyright
Copyright © Royal Society of Edinburgh 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Badiale, M. and Citti, G.. Concentration compactness principle and quasilinear elliptic equations in ℝN. Comm. Partial Differential Equations vol. XLV, (1992) 899–921.Google Scholar
2Badiale, M. and Tchou, N. A.. An existence result for a singular elliptic problem in ℝN (preprint).Google Scholar
3Bahri, A. and Li, Y. Y.. On a min-max procedure for the existence of a positive solution for certain scalar field equations in ℝN. Rev. Mat. Iberoamericana 6 (1990), 1–15.CrossRefGoogle Scholar
4Bahri, A. and Lions, P. L.. On the existence of a positive solution of semilinear elliptic equations in unbounded domains (preprint, CEREMADE 8807).Google Scholar
5Benci, V. and Cerami, G.. Positive solutions of some nonlinear elliptic problems in exterior domains. Arch. Rational Mech. Anal. 99 (1987), 283–300.CrossRefGoogle Scholar
6Benci, V. and Cerami, G.. Existence of positive solutions of the equation -∆u + a(x)u = u(N+2)/(N-2) in ℝN. J. Funct. Anal. 88.1 (1990), 90–118.CrossRefGoogle Scholar
7Berestycki, H. and Lions, P. L.. Nonlinear scalar field equations I. Existence of a ground state. Arch. Rational Mech. Anal. 82 (1983) 313–346.CrossRefGoogle Scholar
8Berestycki, H. and Lions, P. L.. Nonlinear scalar field equations II. Existence of infinitely many solutions. Arch. Rational Mech. Anal. 82 (1983) 347–376.CrossRefGoogle Scholar
9Brezis, H. and Coron, J. M.. Convergence of solutions of H-systems or how to blow bubbles. Arch. Rational Mech. Anal. 89 (1985) 21–56.CrossRefGoogle Scholar
10Brezis, H. and Lieb, E.. A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88 (1983) 486–490.CrossRefGoogle Scholar
11Cao, D. M.. Positive solution and bifurcation from the essential spectrum of a semilinear elliptic equation on ℝN. Nonlinear Anal. 15 (1990), 1045–1052.CrossRefGoogle Scholar
12Capuzzo Dolcetta, I. and Tchou, N. A.. Équations elliptiques semilinéaires avec potentiel singulier. C.R. Acad. Sci. Paris Sér. I 309 (1989), 1007–1012.Google Scholar
13Citti, G.. Positive solutions for a quasilinear degenerate elliptic equation in ℝN. Rend Circ. Mat. Palermo. (2) 35 (1986), 364–375.CrossRefGoogle Scholar
14Citti, G.. Existence of positive solutions of quasilinear degenerate equation on unbounded domains. Ann. Mat. Pura Appl. 183 (1990), 305–330.Google Scholar
15Coffman, C. V. and Marcus, M. M.. Existence theorems for superlinear elliptic Dirichlet problems in exterior domains. Proc. Sympos. Pure Math. 45 (1986), 271–283.CrossRefGoogle Scholar
16Dal Maso, G. and Mosco, U.. Wiener criteria and energy decay for relaxed Dirichlet problems. Arch. Rational Mech. Anal. 95 (4) (1986), 345–387.CrossRefGoogle Scholar
17Dal Maso, G. and Mosco, U.. Wiener criterion and Γ-convergence. Appl. Math. Optimiz. 15 (1987), 15–63.CrossRefGoogle Scholar
18Ding, W. Y. and N, W. N.. On the existence of positive entire solutions of semilinear elliptic equation. Arch. Rational Mech. Anal. 91 (1986), 283–307.CrossRefGoogle Scholar
19Esteban, M. J. and Lions, P. L.. Existence and non existence results for semilinear elliptic problems in unbounded domains. Proc. Roy. Soc. Edinburgh Sect. A 93 (1982), 1–14.CrossRefGoogle Scholar
20Esteban, M. J. and Lions, P. L.. Γ-convergence and the concentration-compactness method for some variational problems with lack of compactness. Ricerche Mat. 36 (1987), 73–101.Google Scholar
21Frehse, J.. Capacity methods in the theory of partial differential equations. Jahresber. Deutsch. Math.-Verein. 84 (1982), 1–44.Google Scholar
22Gidas, B., Ni, W. M. and Nirenberg, L.. Symmetry of positive solutions of nonlinear elliptic equations in ℝN. Adv. Math. Stud. 7A (1981), 369–402.Google Scholar
23Hofer, H.. Variational and topological methods in partially ordered Hilbert spaces. Math. Ann. 261 (1982), 493–514.CrossRefGoogle Scholar
24Kwong, M. K.. Uniqueness of positive solutions of ∆u -u + up = 0 in ℝN. Arch. Rational Mech. Anal. 105 (1989), 243–266.CrossRefGoogle Scholar
25Lions, P. L.. The concentration-compactness principle in the calculus of variations. The locally compact case. Part I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–145.CrossRefGoogle Scholar
26Lions, P. L.. The concentration-compactness principle in the calculus of variations. The locally compact case. Part 2. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223–283.CrossRefGoogle Scholar
27Lions, P. L.. The concentration-compactness principle in the calculus of variations. The limit case. Part I. Rev. Mat. Iberoamericana I. 1 (1985), 145–201.CrossRefGoogle Scholar
28Lions, P. L.. The concentration-compactness principle in the calculus of variations. The limit case. Part 2. Rev. Mat. Iberoamericana II. 1 (1985), 45–121.CrossRefGoogle Scholar
29Lions, P. L.. On positive solutions of semilinear elliptic equations in unbounded domains. In Nonlinear diffusion equations and their equilibrium states, II, eds Ni, W. M., Peletier, L. A. and Serrin, J., 85–122 (Berlin: Springer, 1988).CrossRefGoogle Scholar
30Lions, P. L.. Solutions of Hartree–Fock equations for Coulomb systems. Comm. Math. Phys. 109 (1987), 33–97.CrossRefGoogle Scholar
31Maz'ja, V. G.. Sobolev Spaces (Berlin: Springer, 1985).CrossRefGoogle Scholar
32McLeod, K. and Serrin, J.. Uniqueness of positive radial solutions of ∆u + f(u) = 0 in ℝN. Arch. Rational Mech. Anal. 99/2 (1987), 115–145.CrossRefGoogle Scholar
33Ni, W. M.. Some aspects of semilinear elliptic equations on ℝN. In Nonlinear diffusion equations and their equilibrium states, II, eds Ni, W. M.. Peletier, L. A. and Serrin, J., 171–205 (Berlin: Springer, 1988).CrossRefGoogle Scholar
34Ni, W. M. and Nussbaum, R.. Uniqueness and nonuniqueness for positive radial solutions of ∆u + f(u, r) = 0. Comm. Pure Appl. Math. 38 (1985), 67–108.CrossRefGoogle Scholar
35Rabinowitz, P. H.. Variational methods for nonlinear eigenvalue problems. In Eigenvalues of Nonlinear Problems, C.I.M.E., 141–195 (Rome: Edizioni Cremonese, 1975).Google Scholar
36Strauss, W.. Existence of solitary waves in higher dimensions. Comm. Math Phys. 55 (1977), 149–162.CrossRefGoogle Scholar
37Struwe, M.. A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187 (1984), 511–517.CrossRefGoogle Scholar