Topological Methods in Nonlinear Analysis

Morse theory applied to a $T^{2}$-equivariant problem

Giuseppina Vannella

Full-text: Open access

Abstract

The following $T^2$-equivariant problem of periodic type is considered: $$ \begin{cases} u\in C^2({\mathbb R}^2,{\mathbb R}),\\ -\varepsilon\Delta u(x,y)+F'(u(x,y))=0 & \text{in ${\mathbb R}^{2}$,}\\ u(x,y)=u(x+T,y)=u(x,y+S) &\text{for all $(x,y)\in {\mathbb R}^2$,}\\ \nabla u(x,y)=\nabla u(x+T,y)=\nabla u(x,y+S) &\text{for all $(x,y)\in {\mathbb R}^{2}$.} \end{cases}\tag{\text{P}} $$ Using a suitable version of Morse theory for equivariant problems, it is proved that an arbitrarily great number of orbits of solutions to (P) is founded, choosing $\varepsilon> 0$ suitably small. Each orbit is homeomorphic to $S^1$ or to $T^2$.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 17, Number 1 (2001), 41-53.

Dates
First available in Project Euclid: 22 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1471875796

Mathematical Reviews number (MathSciNet)
MR1846977

Zentralblatt MATH identifier
0992.35035

Citation

Vannella, Giuseppina. Morse theory applied to a $T^{2}$-equivariant problem. Topol. Methods Nonlinear Anal. 17 (2001), no. 1, 41--53. https://projecteuclid.org/euclid.tmna/1471875796


Export citation

References

  • \ref A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications , J. Funct. Anal., 14 (1973), 349–381
  • \ref V. Benci and F. Giannoni, Morse theory for $C^{1}$ functionals and Conley blocks , Topol. Methods Nonlinear Anal., 4 (1994), 365–398
  • \ref V. Benci and F. Pacella, Morse theory for symmetric functionals on the sphere and an application to a bifurcation problem , Nonlinear Anal., 9 (1985), 763–773
  • \ref M. Berger, Nonlinearity and Functional Analysis, Academic Press, London (1977)
  • \ref F. Bethuel, H. Brezis and F. Helein, Ginzburg–Landau Vortices, Birkhäuser (1994)
  • \ref R. Bott, Lectures on Morse theory, old and new , Bull. Amer. Math. Soc., 37 (1988), 331–358
  • \ref G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, London (1972)
  • \ref H. Brezis, Analisi Funzionale, Liguori, Napoli (1986)
  • \ref A. Dold, Lectures on Algebraic Topology, Springer-Verlag, Berlin (1972)
  • \ref M. J. Greenberg and J. R. Harper, Algebraic Topology – a First Course, Addison Wesley, Reedwood City (1981)
  • \ref V. Klee, Some topological properties of convex sets , Trans. Amer. Math. Soc., 78 (1955), 30–45
  • \ref A. Marino and G. Prodi, Metodi perturbativi nella teoria di Morse , Boll. Un. Mat. Ital., 11 \moreref, Suppl. Fasc., 3 (1975), 1–32
  • \ref J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, Berlin (1989)
  • \ref F. Mercuri and G. Palmieri, Morse theory with low differentiability , Boll. Un. Mat. Ital., 7 (1987), 621–631
  • \ref J. Milnor, Morse Theory, Princeton Univ. Press (1963)
  • \ref L. Modica, The gradient theory of phase transitions and the minimal interface criterion , Arch. Rational Mech. Anal., 98 (1986), 123–142
  • \ref L. Modica and S. Mortola, The $\Gamma$-convergence of some functionals , preprint no. 77-7, Istit. Mat. L. Tonelli, Univ. Pisa (1977)
  • \ref R. Palais, Morse theory on Hilbert manifolds , Topology, 2 (1963), 299–340
  • \ref D. Passaseo, Multiplicity of critical points for some functionals related to the minimal surfaces problem , Calc. Var. Partial Differential Equations, 6 (1998), 105–121
  • \ref P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math., 65 , Amer. Math. Soc. (1984)
  • \ref G. Vannella, Some qualitative properties of the solutions of an elliptic equation via Morse theory , Topol. Methods Nonlinear Anal., 9 (1997), 297–312 \ref ––––, Existence and multiplicity of solutions for a nonlinear Neumann problem , Ann. Mat. Pura Appl., to appear