Critical points of a singular perturbation problem via reduced energy and local linking

Journal of Differential Equations

Volumn 159, Issue 2, 1999, Pages 403-426

  Alikakos, Nicholas ^a,b   Kowalczyk, Michał ^c  

^a UNIVERSITY OF TENNESSEE   (United States)

^b UNIVERSITY OF ATHENS   (Greece)

^c CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033544731     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1999.3667     Document Type: Article

References (31)

1. N. D. Alikakos, L. Bronsard, and G. Fusco, Slow Motion in the Gradient Theory of Phase Transitions Via Energy and Spectrum, preprint.
2. N. D. Alikakos and G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions, part I: Spectral estimates, Comm. Part. Diff. Eq. 19 (1994), 1397-1447.
3. N. D. Alikakos and G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions, part II: The motion of bubbles, Arch. Rational Mech Anal., to appear.
4. S. Agmon, "Lectures on Exponential Decay of Solutions of Second-order Elliptic Equations: Bounds on Eigenfunctions of N-body SchröDinger Operators," Princeton University Press, 1982.
5. P. W. Bates and P. Fife, The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl Math., to appear.
6. P. W. Bates and G. Fusco, "nucleating solutions for the Cahn-Hilliard equation in higher space dimension," Lecture Given at the SIAM Conference on Applications of Dynamical Systems, Snowbird, October 15-19, 1992.
7. P. W. Bates and G. Fusco, Equilibria with Many Nuclei for the Cahn-Hilliard Equation, preprint.
8. P. W. Bates, E. N. Dancer and J. Shi, Multi-spike Stationary Solutions of the Cahn-Hilliard Equation in Higher Dimensions and Instability, preprint.
9. V. Benci and P. Rabinowitz, "Critical Point Theorems for Indefinite Functionals," Technical Summary Report #1927, MRC, Univ. of Wisconsin, 1979.
10. D. Cao, N. E. Dancer, E. S. Noussair, and S. Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problem, Disc. Cont. Dyn. Sys. 2(2) (1996), 221-236.
11. M. del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. Henri Poincare, to appear.
12. M. del Pino, P. L. Felmer, and J. Wei, Multi-peak Solutions for Some Singular Perturbation Problems, preprint.
13. A. Gierer and H. Meinhardt, "A Theory of Biological Pattern Formation," Kybernetik, Berlin, 1972, 30-39.
14. C. Gui, Multi-peaked solutions to a semilinear Neumann problem, Duke Math. J. 84 1996, 739-769.
15. C. Gui and J. Wei, Multiple Interior Peak Solutions for Some Singularly Perturbed Neumann Problems, preprint.
16. M. Hirsch, "Differential Topology," Vol. 33, Springer-Verlag, 1976.
17. M. C. Jorge and A. A. Minzoni, A Note on Interface and Bubble Type Asymptotic Solutions to the Two Dimensional Cahn-Hilliard Equation, preprint.
18. M. Kowalczyk, Multiple spike layers in the shadow Gierer-Meinhardt system: Existence of equilibria and quasi-invariant manifold, Duke Math. J., to appear.
19. C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Diff. Eq. 72 (1988), 1-27.
20. W.-M. Ni, Some minimax principles and their applications in nonlinear elliptic equations, J. D'Anal. Math. 37 (1980), 248-275.
21. W.-M. Ni, Diffusion, cross-diffusion and their spike-layer steady states, AMS Notices, to appear.
22. W.-M. Ni and I. Takagi, On the shape of least energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), 819-851.
23. W.-M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), 247-281.
24. W.-M. Ni and I. Takagi, Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan. J. Ind. Appl. Math. 12, 1995, 327-365.
25. W.-M. Ni and H. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problem, Comm. Pure Appl. Math. 48 (1995), 731-768.
26. I. Takagi, Point-condensation for a reaction-diffusion system, J. Diff. Eq. 61 (1986), 208-249.
27. M. Ward, An Asymptotic Analysis of Localized Solutions for Some Reaction-diffusion Models in Multidimensional Domains, preprint.
28. J. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Diff. Eqs. 129 (1996), 315-333.
29. J. Wei, On The Interior Spike Layer Solutions to A Singularly Perturbed Neumann Problem, preprint.
30. J. Wei and M. Winter, Multi-peak Solutions for a Wide Class of Singular Perturbation Problems, preprint.
31. J. Wei and M. Winter, Multi-interior-spike Solutions for the Cahn-Hilliard Equation with Arbitrary Many Peaks, preprint.