Sign-changing solutions of nonlinear elliptic equations

LIU Zhaoli1, WANG Zhi-Qiang2

PDF(216 KB)
PDF(216 KB)
Front. Math. China ›› 2008, Vol. 3 ›› Issue (2) : 221-238. DOI: 10.1007/s11464-008-0014-0

Sign-changing solutions of nonlinear elliptic equations

  • LIU Zhaoli1, WANG Zhi-Qiang2
Author information +
History +

Abstract

In this survey article, we recall some known results on existence and multiplicity of sign-changing solutions of elliptic equations. Methods for obtaining sign-changing solutions developed in the last two decades will also be brief

Cite this article

Download citation ▾
LIU Zhaoli, WANG Zhi-Qiang. Sign-changing solutions of nonlinear elliptic equations. Front. Math. China, 2008, 3(2): 221‒238 https://doi.org/10.1007/s11464-008-0014-0

References

1. Ackermann N Bartsch T Kaplický P et al.Priori bounds, nodal equilibriaand connecting orbits in indefinite superlinear parabolic problemsTrans Amer Math Soc (in press)
2. Aftalion A Pacella F Qualitative properties of nodalsolutions of semilinear elliptic equations in radially symmetric domainsC R Math Acad Sci Paris 2004 339339344
3. Amann H Fixedpoint equations and nonlinear eigenvalue problems in ordered BanachspacesSIAM Rev 1976 18620709. doi:10.1137/1018114
4. Ambrosetti A Rabinowitz P H Dual variational methods incritical point theory and applicationsJFunct Anal 1973 14349381. doi:10.1016/0022‐1236(73)90051‐7
5. Bartsch T Criticalpoint theory on partially ordered Hilbert spacesJ Funct Anal 2001 186117152. doi:10.1006/jfan.2001.3789
6. Bartsch T Chang K -C Wang Z -Q On the Morse indices of sign changing solutions of nonlinearelliptic problemsMath Z 2000 233655677. doi:10.1007/s002090050492
7. Bartsch T Liu Z Multiple sign changing solutionsof a quasilinear elliptic eigenvalue problem involving the p-LaplacianCommContemp Math 2004 6245258. doi:10.1142/S0219199704001306
8. Bartsch T Liu Z On a superlinear elliptic p-Laplacian equationJ Differential Equations 2004 198149175. doi:10.1016/j.jde.2003.08.001
9. Bartsch T Liu Z Weth T Sign changing solutions of superlinear SchrödingerequationsComm Partial Differential Equations 2004 292542. doi:10.1081/PDE‐120028842
10. Bartsch T Liu Z Weth T Nodal solutions of a p-Laplacian equationProc London Math Soc 2005 91129152. doi:10.1112/S0024611504015187
11. Bartsch T Wang Z -Q On the existence of sign changingsolutions for semilinear Dirichlet problemsTopol Methods Nonlinear Anal 1996 7115131
12. Bartsch T Wang Z -Q Sign changing solutions ofnonlinear Schrödinger equationsTopolMethods Nonlinear Anal 1999 13191198
13. Bartsch T Wang Z -Q Willem M The Dirichlet problem for superlinear elliptic equationsIn: Chipot MQuittnerEdsPHandbook of Differential Equations:Stationary Partial Differential Equations, Vol 2AmsterdamElsevier 2005 155
14. Bartsch T Weth T Three nodal solutions of singularlyperturbed elliptic equations on domains without topologyAnn Inst H Poincaré Anal Non Linéaire 2005 22259281. doi:10.1016/j.anihpc.2004.07.005
15. Bartsch T Willem M Infinitely many nonradial solutionsof a Euclidean scalar field equationJ FunctAnal 1993 117447460. doi:10.1006/jfan.1993.1133
16. Bartsch T Willem M Infinitely many radial solutionsof a semilinear elliptic problem on RNArch RationalMech Anal 1993 124261276. doi:10.1007/BF00953069
17. Brezis H Nirenberg L Positive solutions of nonlinearelliptic equations involving critical Sobolev exponentsComm Pure Appl Math 1983 36437477. doi:10.1002/cpa.3160360405
18. Cao D Noussair E S Multiple positive and nodalsolutions for semilinear elliptic problems with critical exponentsIndiana Univ Math J. 1995 4412491271. doi:10.1512/iumj.1995.44.2027
19. Cao D Peng S A note on the sign-changingsolutions to elliptic problems with critical Sobolev and Hardy termsJ Differential Equations 2003 193424434. doi:10.1016/S0022‐0396(03)00118‐9
20. Castro A Cossio J Neuberger J M A sign-changing solution for a superlinear Dirichlet problemRocky Mountain J Math 1997 2710411053
21. Cerami G Solimini S Struwe M Some existence results for superlinear elliptic boundaryvalue problems involving critical exponentsJ Funct Anal 1986 69289306. doi:10.1016/0022‐1236(86)90094‐7
22. Chang K -C A variant mountain pass lemmaSci Sinica,Ser A 1983 2612411255
23. Chang K -C Infinite-dimensional Morse Theory and Multiple Solution Problems.Progress in Nonlinear Differential Equations and Their Applications,No 6BostonBirkhäuser 1993
24. Chang K -C Morse theory in nonlinear analysisIn: Nonlinear Functional Analysis and Applications to Differential Equations(Trieste, 1997)River EdgeWorld Sci Publ 1998 60101
25. Chang K -C Heat method in nonlinear elliptic equationsIn: Topological Methods, Variational Methods and Their Applications (Taiyuan,2002)River EdgeWorld Sci Publ 2003 6576
26. Chang K -C Jiang M Dirichlet problem with indefinitenonlinearitiesCalc Var Partial DifferentialEquations 2004 20257282. doi:10.1007/s00526‐003‐0236‐7
27. Conti M Merizzi L Terracini S Remarks on variational methods and lower-upper solutionsNonlinear Differential Equations Appl 1999 6371393. doi:10.1007/s000300050009
28. Conti M Terracini S Verzini G Nehari's problem and competing species systemsAnn Inst H Poincaré Anal Non Linéaire 2002 19871888. doi:10.1016/S0294‐1449(02)00104‐X
29. Costa D Wang Z -Q Multiplicity results for aclass of superlinear elliptic problemsProcAmer Math Soc 2005 133787794. doi:10.1090/S0002‐9939‐04‐07635‐X
30. Dancer E Du Y Competing species equationswith diffusion, large interaction, and jumping nonlinearitiesJ Differential Equations 1994 114434475. doi:10.1006/jdeq.1994.1156
31. Dancer E Du Y On sign-changing solutionsof certain semilinear elliptic problemsAppl Anal 1995 56193206. doi:10.1080/00036819508840321
32. Dancer E Du Y Multiple solutions of somesemilinear elliptic equations via the generalied Conley indexJ Math Anal Appl 1995 189848871. doi:10.1006/jmaa.1995.1054
33. Dancer E Du Y A note on multiple solutionsof some semilinear elliptic problemsJ MathAnal Appl 1997 211626640. doi:10.1006/jmaa.1997.5471
34. Dancer E Wei J Sign-changing solutions forsupercritical elliptic problems in domains with small holesManuscripta Math 2007 123493511. doi:10.1007/s00229‐007‐0110‐6
35. Hofer H Variationaland topological methods in partially ordered Hilbert spacesMath Ann 1982 261493514. doi:10.1007/BF01457453
36. Jiang M Criticalgroups and multiple solutions of the p-Laplacian equationsNonlinear Anal 2004 5912211241
37. Li C Li S Multiple solutions and sign-changingsolutions of a class of nonlinear elliptic equations with Neumannboundary conditionJ Math Anal Appl 2004 2981432. doi:10.1016/j.jmaa.2004.01.017
38. Li S Wang Z -Q Mountain pass theorem in orderintervals and multiple solutions for semilinear elliptic DirichletproblemsJ Anal Math 2000 81373396
39. Li S Wang Z -Q Ljusternik-Schnirelman theoryin partially ordered Hilbert spacesTransAmer Math Soc 2002 35432073227. doi:10.1090/S0002‐9947‐02‐03031‐3
40. Li S Zhang Z Sign-changing solution andmultiple solutions theorems for semilinear elliptic boundary valueproblems with jumping nonlinearitiesActaMath Sinica 2001 44507516
41. Li Y Liu Z Multiple and sign changingsolutions of an elliptic eigenvalue problem with constraintSci China, Ser A 2001 444857. doi:10.1007/BF02872282
42. Liu J Wang Y Wang Z -Q Solutions for quasilinear Schrödinger equations viathe Nehari methodComm Partial DifferentialEquations 2004 29879901. doi:10.1081/PDE‐120037335
43. Liu Z MultipleSolutions of Differential EquationsPh D Thesis JinanShandong Univ 1992
44. Liu Z Li Y Solutions of an elliptic eigenvalueproblem involving subcritical or critical exponentsComm Partial Differential Equations 2001 2622272248. doi:10.1081/PDE‐100107820
45. Liu Z Sun J Invariant sets of descendingflow in critical point theory with applications to nonlinear differentialequationsJ Differential Equations 2001 172257299. doi:10.1006/jdeq.2000.3867
46. Liu Z Sun J Number of invariant sets ofdescending flow with applications in critical point theoryIn: Brezis HLi S JLiu J Q et al.MorseTheory, Minimax Theory and Their Applications to Nonlinear DifferentialEquationsBostonInt Press 2003 139156
47. Liu Z Sun J Four versus two solutions ofsemilinear elliptic boundary value problemsCalc Var Partial Differential Equations 2002 14319327. doi:10.1007/s005260100104
48. Liu Z van Heerden F A Wang Z -Q Nodal type bound states of Schrödinger equations viainvariant set and minimax methodsJ DifferentialEquations 2005 214358390. doi:10.1016/j.jde.2004.08.023
49. Liu Z Wang Z -Q On the Ambrosetti-Rabinowitzsuperlinear conditionAdv Nonlinear Stud 2004 4563574
50. Liu Z Wang Z -Q Schrödinger equationswith concave and convex nonlinearitiesZAngew Math Phys 2005 56609629. doi:10.1007/s00033‐005‐3115‐6
51. Liu Z Wang Z -Q Multi-bump type nodal solutionshaving a prescribed number of nodal domains. IAnn Inst H Poincaré Anal Non Linéaire 2005 22597608. doi:10.1016/j.anihpc.2004.10.002
52. Liu Z Wang Z Multi-bump type nodal solutionshaving a prescribed number of nodal domains. IIAnn Inst H Poincaré Anal Non Linéaire 2005 22609631. doi:10.1016/j.anihpc.2004.10.003
53. Liu Z Wang Z -Q Weth T Multiple solutions of nonlinear Schrödinger equationsvia flow invariance and Morse theoryProcRoy Soc Edinburgh Sect A 2006 136945969. doi:10.1017/S0308210500004820
54. Rabinowitz P H MinimaxMethods in Critical Point Theory with Applications to DifferentialEquationsCBMS Conf Ser in Math, No 65ProvidenceAmerMath Soc 1986
55. Rabinowitz P H Su J Wang Z -Q Multiple solutions of a superlinear elliptic equationRend Lincei di Matematica 2007 1897108
56. Schecheter M Wang Z -Q Zou W New linking theorem and sign-changing solutionsComm Partial Differential Equations 2004 29471488. doi:10.1081/PDE‐120030405
57. Schechter M Zou W Infinitely many solutions toperturbed elliptic equationsJ Funct Anal 2005 228138. doi:10.1016/j.jfa.2005.06.014
58. Schechter M Zou W Sign-changing critical pointsfrom linking type theoremsTrans Amer MathSoc 2006 35852935318. doi:10.1090/S0002‐9947‐06‐03852‐9
59. Sun J Topicson Nonlinear OperatorsPh D Thesis JinanShandong Univ 1984
60. Sun J TheSchauder condition in the critical point theoryKexue Tongbao 1986 3111571162
61. Sun J Liu Z Calculus of variations andsuper-and sub-solutions in reverse orderActa Math Sinica 1994 37512514
62. Wang Z -Q On a superlinear elliptic equationAnnInst H Poincaré Anal Non Linéaire 1991 84357
63. Wang Z -Q Nonlinear boundary value problems with concave nonlinearities nearthe originNonlinear Differential EquationsAppl 2001 81533. doi:10.1007/PL00001436
64. Wang Z -Q Minimax methods, invariant sets, and applications to nodal solutionsof nonlinear elliptic problemsIn: Proceedingsof EquaDiff 03, Hasselt 2003SingaporeWorld Scientific 2005 561566
65. Wang Z -Q Zhou J A local minimax-Newton methodfor finding multiple saddle points with symmetriesSIAM J Numer Anal 2004 4217451759. doi:10.1137/S0036142903431675
66. Wang Z -Q Zhou J An efficient and stable methodfor computing multiple saddle points with symmetriesSIAM J Numer Anal 2005 43891907. doi:10.1137/S0036142903416626
67. Zhang Z Li S On sign-changing and multiplesolutions of the p-LaplacianJ Funct Anal 2003 197447468. doi:10.1016/S0022‐1236(02)00103‐9
68. Zou W Sign-changingsaddle pointJ Funct Anal 2005 219433468
69. Zou W Onfinding sign-changing solutionsJ FunctAnal 2006 234364419. doi:10.1016/j.jfa.2005.09.004
AI Summary AI Mindmap
PDF(216 KB)

Accesses

Citations

Detail

Sections
Recommended

/