Symmetry of nodal solutions for singularly perturbed elliptic problems on a ball

Indiana University Mathematics Journal

Volumn 54, Issue 3, 2005, Pages 707-742

  Wei, Juncheng ^a   Winter, Matthias ^b  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^b UNIVERSITY OF STUTTGART   (Germany)

Author keywords

Liapunov Schmidt reduction; Method of moving planes; Singular perturbation problems; Symmetry of nodal solutions

Indexed keywords

EID: 23244437368     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2005.54.2546     Document Type: Article

References (57)

1. NICHOLAS D. ALIKAKOS, GIORGIO FUSCO, and MICHAŁ KOWALCZYK, Finite-dimensional dynamics and interfaces intersecting the boundary: equilibria and quasi-invariant manifold, Indiana Univ. Math. J. 45 (1996), 1119-1155, http://dx.doi.org/10.1512/iumj.1996.45.1123. MR1444480 (98b:35077)
2. NICHOLAS D. ALIKAKOS and MICHAŁ KOWALCZYK, Critical points of a singular perturbation problem via reduced energy and local linking, J. Differential Equations 159 (1999), 403-426, http://dx.doi.org/10.1006/jdeq.1999. 3667. MR1730726 (2001b:35034)
3. PETER W. BATES, NORMAN E. DANCER, and JUNPING SHI, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Differential Equations 4 (1999), 1-69. MR1667283 (99k:35097)
4. PETER W. BATES and GIORGIO FUSCO, Equilibria with many nuclei for the Cahn-Hilliard equation, J. Differential Equations 160 (2000), 283-356, http://dx.doi.org/10.1006/jdeq.1999.3660. MR1737000 (2001b:35l45)
5. PETER W. BATES and JUNPING SHI, Existence and instability of spike layer solutions to singular perturbation problems, J. Funct. Anal. 196 (2002), 211-264, http://dx.doi.org/10.1016/S0022-1236(02)00013-7. MR1943093 (2003k:35045)
6. VIERI BENCI and GIOVANNA CERAMI, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal. 99 (1987), 283-300, http://dx.doi.org/10.1007/BF00282048. MR898712 (88f:5014)
7. N, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 567-588. MR1409663 (97k:35058)
8. DAOMIN CAO, NORMAN E. DANCER, EZZAT S. NOUSSAIR, and SHUNSEN YAN, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems, Discrete Contin. Dynam. Systems 2 (1996), 221-236. MR1382508 (96m:35095)
9. GIOVANNA CERAMI and JUNCHENG WEI, Multiplicity of multiple interior peak solutions for some singularly perturbed Neumann problems, Internat. Math. Res. Notices (1998), 601-626, http://dx.doi.org/10.11155/S1073792898000385. MR1635869 (99j:35063)
10. CHUIN CHUAN CHEN and CHANG SHOU LIN, On the symmetry of blowup solutions to a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 271-296, http://dx.doi.org/10.1016/S0294-1449(00) 00060-3. MR1831657 (2002d:35064) (English, with English and French summaries)
11. NORMAN E. DANCER, The effect of domain shape on the number of positive solutions of certain nonlinear equations. II, J. Differential Equations 87 (1990), 316-339, http://dx.doi.org/10.1016/0022-0396(90)90005-A. MR1072904 (91j:35034)
12. NORMAN E. DANCER and JUNCHENG WEI, On the effect of domain topology in a singular perturbation problem, Topol. Methods Nonlinear Anal. 11 (1998), 227-248. MR1659466 (2000a:35012)
13. NORMAN E. DANCER and SHUSEN YAN, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math. 189 (1999), 241-262. MR1696122 (2000d:35010)
14. _, Interior and boundary peak solutions for a mixed boundary value problem, Indiana Univ. Math. J. 48 (1999), 1177-1212, http://dx.doi.org/10.1512/ iumj.1999.48.1827. MR1757072 (2001B5146)
15. MANUEL DEL PINO, PATRICIO L. FELMER, and JUNCHENG WEI, Multi-peak solutions for some singular perturbation problems, Calc. Var. Partial Differential Equations 10 (2000), 119-134, http://dx.doi.org/10.1007/ s005260050147. MR1750734 (2001a:35019)
16. _, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal. 31 (1999), 63-79 (electronic), http://dx.doi.org/10.1137/S0036141098332834. MR1742305 (2000k:35103)
17. _, On the role of distance function in some singular perturbation problems, Comm. Partial Differential Equations 25 (2000), 155-177, http://dx.doi.org/10.1007/s005260050147. MR1737546 (2000m:35017)
18. ANDREAS FLOER and ALAN WEINSTEIN, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986), 397-408, http://dx.doi.org/10.1016/0022-1236(86)90096-0. MR867665 (88d:35169)
19. BASILIS GIDAS, WEI MING NI, and LOUIS NIRENBERG, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243, http://dx.doi.org/10.1007/BF01221125. MR544879 (80h:35043)
20. n, Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York, 1981, pp. 369-402. MR634248 (84a:35083)
21. ALFRED GIERER and HANS MEINHARDT, A theory of biological pattern formation, Kybernetik 12 (1972), 30-39, http://dx.doi.org/10.1007/BF00289234.
22. MASSIMO GROSSI, Uniqueness of the least-energy solution for a semilinear Neumann problem, Proc. Amer. Math. Soc. 128 (2000), 1665-1672, http://dx.doi.org/10.1090/S0002-9939-99-05491-X. MR1694340(2000j:35097)
23. MASSIMO GROSSI, ANGELA PISTOIA, and JUNCHENG WEI, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Calc. Var. Partial Differential Equations 11 (2000), 143-175. MR1782991 (2001h:35012)
24. CHANGFENG GUI and JUNCHENG WEI, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations 158 (1999), 1-27, http://dx.doi.org/10.1016/S0022-0396(99)80016-3. MR1721719 (2000g:35035)
25. _, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math. 52 (2000), 522-538. MR1758231 (2001b:35023)
26. CHANGFENG GUI, JUNCHENG WEI, and MATTHIAS WINTER, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), 47-82, http://dx.doi.org/ 10.1016/S0294-1449(99)00104-3. MR1743431 (2001a:35018) (English, with English and French summaries)
27. MICHAŁ KOWALCZYK, Multiple spike layers in the shadow Gierer-Meinhardt system: existence of equilibria and the quasi-invariant manifold, Duke Math. J. 98 (1999), 59-111, http://dx.doi.org/10.1215/S0012-7094- 99-09802-2. MR1687412 (2000d:35073)
28. n, Arch. Rational Mech. Anal. 105 (1989), 243-266, http://dx.doi.org/10.1007/BF00251502. MR969899 (90d:35015)
29. VAN YAN LI, On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differential Equations 23 (1998), 487-545. MR1620632 (2000a:35013)
30. YAN YAN LI and LOUIS NIRENBERG, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math. 51 (1998), 1445-1490, http://dx.doi.org/10.1002/(SICI)1097-0312(199811/12)51:11/12¡1445:: AID-CPA9¿3.0.CO;2-Z. MR1639159 (99g:35014)
31. CHANG SHOU LIN, Locating the peaks of solutions via the maximum principle. I. The Neumann problem, Comm. Pure Appl. Math. 54 (2001), 1065-1095, http://dx.doi.org/10.1002/cpa.1017. MR1835382 (2002d:35052)
32. CHANG SHOU LIN and IZUMI TAKAGI, Method of rotating planes applied to a singularly perturbed Neumann problem, Calc. Var. Partial Differential Equations 13 (2001), 519-536, http://dx.doi.org/10.1007/s005260100087. MR1867940 (2002k:35116)
33. CHANG SHOU LIN and WEI MING NI, On the diffusion coefficient of a semilinear Neumann problem, Calculus of Variations and Partial Differential Equations (Trento, 1986), Lecture Notes in Math., vol. 1340, Springer, Berlin, 1988, pp. 160-174. MR974610 (90d:35101)
34. CHANG SHOU LIN and JUNCHENG WEI, Uniqueness of multiple-spike solutions via the method of moving planes (submitted).
35. WEI MING NI and IZUMI TAKAGI, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), 819-851. MR1115095 (92i:35052)
36. _, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), 247-281, http://dx.doi.org/10.1215/S0012-7094- 93-07004-4. MR1219814 (94h:35072)
37. WEI MING NI and JUNCHENG WEI, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math. 48 (1995), 731-768. MR1342381 (96g:35077)
38. WEI MING NI, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1998), 9-18. MR1490535 (99a:35132)
39. EZZAT S. NOUSSAIR, On the number of nodal solutions in singular perturbation problems, Differential Integral Equations 14 (2001), 1093-1109. MR1852873 (2002f:5019)
40. EZZAT S. NOUSSAIR and SHUSEN YAN, The effect of the domain geometry in singular perturbation problems, Proc. London Math. Soc. (3) 76 (1998), 427-452, http://dx.doi.org/10.1112/S0024611598000148. MR1490244 (98m:35013)
41. EZZAT S. NOUSSAIR and JUNCHENG WEI, On the effect of domain geometry on the existence of nodal solutions in singular perturbations problems, Indiana Univ. Math. J. 46 (1997), 1255-1271, http://dx.doi.org/10.1512/iumj.1997.46. 1401. MR1631584 (99e:35011)
42. _, On the location of spikes and profile of nodal solutions for a singularly perturbed Neumann problem, Comm. Partial Differential Equations 23 (1998), 793-816. MR1632748 (99h:35013)
43. a, Comm. Partial Differential Equations 13 (1988), 1499-1519. MR970154(90d:35063a)
44. _, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys. 131 (1990), 223-253. MR1065671 (92a:35148)
45. OLIVIER REV, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990), 1-52, http://dx.doi.org/10.1016/0022-1236(90)90002-3. MR1040954 (91b:35012)
46. MICHAEL J. WARD, An asymptotic analysis of localized solutions for some reaction-diffusion models in multidimensional domains, Stud. Appl. Math. 97 (1996), 103-126. MR1395845 (97c:35102)
47. JUNCHENG WEI, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations 129 (1996), 315-333, http://dx.doi.org/10.1006/jdeq.1996.0120. MR1404386 (97f:5015)
48. _, On the boundary spike layer solutions to a singularly perturbed Neumann problem, J. Differential Equations 134 (1997), 104-133, http://dx.doi.org/10.1006/jdeq.1996.3218. MR1429093 (98e:35076)
49. _, Conditions for two-peaked solutions of singularly perturbed elliptic equations, Manuscripta Math. 96 (1998), 113-131, http://dx.doi.org/10.1007/ s002290050057. MR1624364 (99e:35014)
50. _, On the interior spike solutions for some singular perturbation problems, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 849-874. MR1635448 (99h:35012)
51. _, On the interior spike layer solutions to a singularly perturbed Neumann problem, Tohoku Math. J. (2) 50 (1998), 159-178. MR1622042 (99g:35016)
52. _, Uniqueness and critical spectrum of boundary spike solutions, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 1457-1480. MR1869645 (2002j:35023)
53. _, On the effect of domain geometry in singular perturbation problems, Differential Integral Equations 13 (2000), 15-45. MR1811947 (2002c:35032)
54. _, On single interior spike solutions of the Gierer-Meinhardt system: uniqueness and spectrum estimates, European J. Appl. Math. 10 (1999), 353-378, http://dx.doi.org/10.1017/S0956792599003770. MR1713076 (2001c:92003)
55. _, Point-condensation generated by the Gierer-Meinhardt system: a brief survey, New Trends In Partial Differential Equations (Y. Morita, H. Ninomiya, E. Yanagida, and S. Yotsutani, eds.), Ryukoku University, Ryukoku, 2000, pp. 46-59.
56. JUNCHENG WEI and MATTHIAS WINTER, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 459-492, http://dx.doi.org/10.1016/S0294-1449(98)80031-0. MR1632937 (2000b:35093) (English, with English and French summaries)
57. _, Multi-peak solutions for a wide class of singular perturbation problems, J. London Math. Soc. (2) 59 (1999), 585-606, http://dx.doi.org/10. 1112/S002461079900719X. MR1709667 (2000h:35012)