Arkiv för Matematik

  • Ark. Mat.
  • Volume 41, Number 1 (2003), 105-114.

Infinitely many solutions of a symmetric semilinear elliptic equation on an unbounded domain

Sara Maad

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Abstract

We study a semilinear elliptic equation of the form $ - \Delta u + u = f(x,u), u \in H_0^1 (\Omega ),$ where f is continuous, odd in u and satisfies some (subcritical) growth conditions. The domain Ω⊂RN is supposed to be an unbounded domain (N≥3). We introduce a class of domains, called strongly asymptotically contractive, and show that for such domains Ω, the equation has infinitely many solutions.

Article information

Source
Ark. Mat., Volume 41, Number 1 (2003), 105-114.

Dates
Received: 17 July 2001
Revised: 19 May 2002
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898794

Digital Object Identifier
doi:10.1007/BF02384570

Mathematical Reviews number (MathSciNet)
MR1971943

Zentralblatt MATH identifier
1088.35019

Rights
2003 © Institut Mittag-Leffler

Citation

Maad, Sara. Infinitely many solutions of a symmetric semilinear elliptic equation on an unbounded domain. Ark. Mat. 41 (2003), no. 1, 105--114. doi:10.1007/BF02384570. https://projecteuclid.org/euclid.afm/1485898794


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References

  • Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.
  • Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, 1998.
  • Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–145.
  • Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223–283.
  • Del Pino, M. A. and Felmer, P. H., Least energy solutions for elliptic equations in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 195–208.
  • Schindler, I. and Tintarev, K., Semilinear elliptic problems on unbounded domains, in Calculus of Variations and Differential Equations (Haifa, 1998) (Ioffe, A., Reich, S. and Shafrir, I., eds.), pp. 210–217. Chapman & Hall/CRC, Boca Raton, Fla., 2000.
  • Schindler, I. and Tintarev, K., An abstract version of the concentration compactness principle, to appear in Rev. Mat. Univ. Complut. Madrid.
  • Struwe, M., Variational Methods. 2nd ed., Springer-Verlag, Berlin. 1996.
  • Tintarev, K., Solutions to elliptic systems of Hamiltonian type in RN. Electron. J. Differential Equations 29 (1999). 1–11.
  • Willem, M., Minimax Theorems, Birkhäuser, Boston, Mass., 1996.