Boundary Concentration Phenomena for a Singularly Perturbed Elliptic Problem

Communications on Pure and Applied Mathematics

Volumn 55, Issue 12, 2002, Pages 1507-1568

  Malchiodi, Andrea ^a   Montenegro, Marcelo ^b  

^a INSTITUTE FOR ADVANCED STUDY   (United States)

^b UNIVERSITY OF CAMPINAS   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036425683     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.10049     Document Type: Article

References (42)

1. Ambrosetti, A.; Badiale, M. Homoclinics: Poincaré-Melnikov type results via a variational approach. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), no. 2, 233-252.
2. Ambrosetti, A.; Badiale, M.; Cingolani, S. Semiclassical states of nonlinear Schrödmger equations. Arch. Rational Mech. Anal. 140 (1997), no. 3, 285-300.
3. Ambrosetti, A.; Malchiodi, A.; Secchi, S. Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal. 159 (2001), no. 3, 253-271.
4. Badiale, M,; D'Aprile, T. Concentration around a sphere for a singularly perturbed Schrödinger equation. Nonlinear Anal. 49 (2002), no. 7, 947-985.
5. Bates, P.; Fusco, G. Multi-spike states of the Cahn-Hilliard model for phase transitions. Preprint, 2001.
6. Berestycki, H.; Lions, P.-L. Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82 (1983), no. 4, 313-345.
7. n, n ≥ 3. Comm. Partial Differential Equations 16 (1991), no. 8-9, 1549-1572.
8. Dancer, E. N.; Wei, J. On the effect of domain topology in a singular perturbation problem. Topol. Methods Nonlinear Anal. 11 (1998), no. 2, 227-248.
9. Dancer, E. N.; Yan, S. Multipeak solutions for a singularly perturbed Neumann problem. Pacific J. Math. 189 (1999), no. 2, 241-262.
10. del Pino, M.; Felmer, P. L. Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differential Equations 4 (1996), no. 2, 121-137.
11. del Pino, M.; Felmer, P. Semi-classcal states for nonlinear Schrödinger equations. J. Funct. Anal. 149 (1997), no. 1, 245-265.
12. del Pino, M.; Felmer, P. L.; Wei, J. On the role of the mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal. 31 (1999), no. 1, 63-79.
13. Floer, A.; Weinstein, A. Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69 (1986), no. 3, 397-408.
14. Gidas, B.; Ni, W. M.; Nirenberg, L. Symmetry of positive solutions of nonlinear elliptic equations in Rn. Mathematical analysis and applications, Part A, 369-402. Advances in Mathematics Supplementary Studies, 7A. Academic, New York-London, 1981.
15. Gierer, A.; Meinhardt, H. A theory of biological pattern formation. Kybernetik 12 (1972), 30-39.
16. Gilbarg, D.; Trudinger, N. S. Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften, 224. Springer, Berlin, 1983.
17. Grossi, M. Some results on a class of nonlinear Schrödinger equations. Math. Z. 235 (2000), no. 4, 687-705.
18. Grossi, M.; Pistoia, A.; Wei, J. Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory. Calc. Var. Partial Differential Equations 11 (2000), no. 2, 143-175.
19. Gui, C. Multipeak solutions for a semilinear Neumann problem. Duke Math. J. 84 (1996), no. 3, 739-769.
20. Gui, C.; Wei, J. Multiple interior peak solutions for some singularly perturbed Neumann problems. J. Differential Equations 158 (1999), no. 1, 1-27.
21. Gui, C.; Wei, J. On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems. Canad. J. Math. 52 (2000), no. 3, 522-538.
22. Gui, C.; Wei, J.; Winter, M. Multiple boundary peak solutions for some singularly perturbed Neumann problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 1, 47-82.
23. Kato, T. Perturbation theory for linear operators. Second edition. Grundlehren der Mathematischen Wissenschaften, 132. Springer, Berlin-New York, 1976.
24. Keller, E. F.; Segal, L. A. Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970), 399-415.
25. n. Arch. Rational Mech. Anal 105 (1989), no. 3, 243-266.
26. Li, Y. Y. On a singularly perturbed equation with Neumann boundary condition. Comm. Partial Differential Equations 23 (1998), no. 34, 487-545.
27. Li, Y.; Nirenberg, L. The Dirichlet problem for singularly perturbed elliptic equations. Comm. Pure Appl. Math. 51 (1998), no. 11-12, 1445-1490.
28. Lin, C.-S.; Ni, W.-M.; Takagi, I. Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations 72 (1988), no. 1, 1-27.
29. n. Asymptot. Anal. 25(2001), no. 2, 149-181.
30. Ni, W.-M. Diffusion, cross-diffusion, and their spike-layer steady states. Notices Amer. Math. Soc. 45 (1998), no. 1, 9-18.
31. Ni, W.-M.; Pan, X. B.; Takagi, I. Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents. Duke Math. J. 67 (1992), no. 1, 1-20.
32. Ni, W.-M.; Takagi, I. On the shape of least-energy solutions to a semilinear Neumann problem. Comm. Pure Appl. Math. 41 (1991), no. 7, 819-851.
33. Ni, W.-M.; Takagi, I. Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 70 (1993), no. 2, 247-281.
34. Ni, W.-M.; Takagi, I.; Wei, J. On the location and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem: intermediate solutions. Duke Math. J. 94 (1998), no. 3, 597-618.
35. Ni, W.-M.; Takagi, L; Yanagida, E. Stability of least energy patterns of the shadow system for an activator-inhibitor model. Recent topics in mathematics moving toward science and engineering. Japan J. Indust. Appl. Math. 18 (2001), no. 2, 259-272.
36. Ni, W. M.; Wei, J. On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. Comm. Pure Appl. Math. 48 (1995), no. 7, 731-768.
37. Oh, Y. G. On positive multibump states of nonlinear Schrödinger equation under multiple well potentials. Comm. Math. Phys. 131 (1990), no. 2, 223-253.
38. Shatah, J.; Zeng, C. Periodic solutions for Hamiltonian systems under strong constraining forces. Preprint, 2001.
39. Wang, Z. Q. On the existence of multiple, single-peaked solutions for a semilinear Neumann problem. Arch. Rational Mech. Anal. 120 (1992), no. 4, 375-399.
40. Wei, J. On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem. J. Differential Equations 129 (1996), no. 2, 315-333.
41. Wei, J. On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem. J. Differential Equations 134 (1997), no. 1, 104-133.
42. Zhang, L. Q. Uniqueness of ground state solutions, Acta Math. Sci. (English ed.) 8 (1988), no. 4, 449-468.