November  2019, 12(7): 1807-1833. doi: 10.3934/dcdss.2019119

The work of Norman Dancer

School of Science and Technology, University of New England, Armidale, NSW 2351, Australia

Prof. Norman Dancer's help during the preparation of this article is gratefully acknowledged. The author also thanks the editors for advices leading to an improved version of this article

Received  June 2018 Revised  July 2018 Published  December 2018

In this article, a sample of Norman Dancer's published works are briefly described, to give the reader a taste of his deep and important research on nonlinear functional analysis, nonlinear ODE and PDE problems, and dynamical systems. The sample covers a variety of topics where Norman Dancer has made remarkable contributions.

The author takes this opportunity to express his deep admiration of the work of Professor Norman Dancer, and to thank him for the kind help to the development of the author's career, which has been greatly influenced by him and his work. 

Citation: Yihong Du. The work of Norman Dancer. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1807-1833. doi: 10.3934/dcdss.2019119
References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.

[2]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl.(4), 93 (1972), 231-246. doi: 10.1007/BF02412022.

[3]

L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $\mathbb R^3$ and a conjecture of De Giorgi, J. Am. Math. Soc., 13 (2000), 725-739.  doi: 10.1090/S0894-0347-00-00345-3.

[4]

A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294.  doi: 10.1002/cpa.3160410302.

[5]

Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, Springfield, MO, 2006.

[6]

K. Borsuk, Drei Sätze über die $n$-dimensionale Sphäre, Fund. Math., 20 (1933), 236-243.

[7]

B. BuffoniE. N. Dancer and J. F. Toland, The regularity and local bifurcation of steady periodic water waves, Arch. Ration. Mech. Anal., 152 (2000), 207-240.  doi: 10.1007/s002050000086.

[8]

B. BuffoniE. N. Dancer and J. F. Toland, The sub-harmonic bifurcation of Stokes waves, Arch. Ration. Mech. Anal., 152 (2000), 241-271.  doi: 10.1007/s002050000087.

[9]

D. Clark, A variant of the Lyusternik-Schnirelmann theory, Indiana Univ. Math. J., 22 (1973), 65-74.  doi: 10.1512/iumj.1973.22.22008.

[10]

C. Conley, Isolated Invariant Sets and The Morse Index, CBMS regional conference series in mathematics No. 38., Providence: Amer. Math. Soc., 1978.

[11]

M. ContiS. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560.  doi: 10.1016/j.aim.2004.08.006.

[12]

E. C. M. Crooks, E. N. Dancer and D. Hilhorst, On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions, Topol. Methods Nonlinear Anal., 30 (2007), 1-36.

[13]

M. CuestaD. de Figueiredo and J.-P. Gossez, The beginning of the Fucik Spectrum for the p-Laplacian, J. Differ. Equ., 159 (1999), 212-238.  doi: 10.1006/jdeq.1999.3645.

[14]

E. N. Dancer, Bifurcation theory in real Banach space, Proc. London Math. Soc., (3) 23 (1971), 699-734. doi: 10.1112/plms/s3-23.4.699.

[15]

E. N. Dancer, Bifurcation theory for analytic operators, Proc. London Math. Soc., (3) 26 (1973), 359-384. doi: 10.1112/plms/s3-26.2.359.

[16]

E. N. Dancer, Global structure of the solutions of non-linear real analytic eigenvalue problems, Proc. London Math. Soc., (3) 27 (1973), 747-765. doi: 10.1112/plms/s3-27.4.747.

[17]

E. N. Dancer, On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 76 (1977), 283-300.  doi: 10.1017/S0308210500019648.

[18]

E. N. Dancer, On the existence of solutions of certain asymptotically homogeneous problems, Math. Z., 177 (1981), 33-48.  doi: 10.1007/BF01214337.

[19]

E. N. Dancer, Symmetries, degree, homotopy indices and asymptotically homogeneous problems, Nonlinear Anal., 6 (1982), 667-686.  doi: 10.1016/0362-546X(82)90037-2.

[20]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.  doi: 10.1016/0022-247X(83)90098-7.

[21]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.  doi: 10.1090/S0002-9947-1984-0743741-4.

[22]

E. N. Dancer, On positive solutions of some pairs of differential equations, II, J. Differential Equations, 60 (1985), 236-258.  doi: 10.1016/0022-0396(85)90115-9.

[23]

E. N. Dancer, A new degree for $S^1$-invariant gradient mappings and applications, Ann. Inst. H. Poincare Anal. Non Linaire, 2 (1985), 329-370.  doi: 10.1016/S0294-1449(16)30396-1.

[24]

E. N. Dancer, Counterexamples to some conjectures on the number of solutions of nonlinear equations, Math. Ann., 272 (1985), 421-440.  doi: 10.1007/BF01455568.

[25]

E. N. Dancer, Multiple fixed points of positive mappings, J. Reine Angew. Math., 371 (1986), 46-66.  doi: 10.1515/crll.1986.371.46.

[26]

E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations, 74 (1988), 120-156.  doi: 10.1016/0022-0396(88)90021-6.

[27]

E. N. Dancer, On the influence of domain shape on the existence of large solutions of some superlinear problems, Math. Ann., 285 (1989), 647-669.  doi: 10.1007/BF01452052.

[28]

E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, Ⅱ, J. Differential Equations, 87 (1990), 316-339.  doi: 10.1016/0022-0396(90)90005-A.

[29]

E. N. Dancer and J. F. Toland, Degree theory for orbits of prescribed period of flows with the first integral, Proc. London Math. Soc., 60 (1990), 549-580.  doi: 10.1112/plms/s3-60.3.549.

[30]

E. N. Dancer and P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems, J. Reine Angew. Math., 419 (1991), 125-139.

[31]

E. N. Dancer, Generic domain dependence for nonsmooth equations and the open set problem for jumping nonlinearities, Topol. Methods Nonlinear Anal., 1 (1993), 139-150.  doi: 10.12775/TMNA.1993.011.

[32]

E. N. Dancer and Y. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations, 114 (1994), 434-475.  doi: 10.1006/jdeq.1994.1156.

[33]

E. N. Dancer and Y. Du, Existence of changing sign solutions for some semilinear problems with jumping nonlinearities at zero, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 1165-1176.  doi: 10.1017/S0308210500030171.

[34]

E. N. Dancer and P. Hess, Stable subharmonic solutions in periodic reaction-diffusion equations, J. Differential Equations, 108 (1994), 190-200.  doi: 10.1006/jdeq.1994.1032.

[35]

E. N. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion. Ⅰ, General existence results. Nonlinear Anal., 24 (1995), 337-357. doi: 10.1016/0362-546X(94)E0063-M.

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E. N. Dancer, Domain variation for certain sets of solutions and applications, Top. Meth. Nonl. Anal., 7 (1996), 95-113.  doi: 10.12775/TMNA.1996.004.

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E. N. Dancer and D. Daners, Domain perturbation for elliptic equations subject to Robin boundary conditions, J. Differential Equations, 138 (1997), 86-132.  doi: 10.1006/jdeq.1997.3256.

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E. N. DancerD. HilhorstM. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system, European J. Appl. Math., 10 (1999), 97-115.  doi: 10.1017/S0956792598003660.

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E. N. Dancer and K. Perera, Some remarks on the Fucik spectrum of the p-Laplacian and critical groups, J. Math. Anal. Appl., 254 (2001), 164-177.  doi: 10.1006/jmaa.2000.7228.

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E. N. Dancer and Z. Zhang, Dynamics of Lotka-Volterra competition systems with large interaction, J. Differential Equations, 182 (2002), 470-489.  doi: 10.1006/jdeq.2001.4102.

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E. N. Dancer, Real analyticity and non-degeneracy, Math. Ann., 325 (2003), 369-392.  doi: 10.1007/s00208-002-0352-2.

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E. N. Dancer, Stable and finite Morse index solutions on $R^n$ or on bounded domains with small diffusion, Ⅱ, Indiana Univ. Math. J., 53 (2004), 97-108.  doi: 10.1512/iumj.2004.53.2354.

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E. N. Dancer, Stable and finite Morse index solutions on $R^n$ or on bounded domains with small diffusion, Trans. Amer. Math. Soc., 357 (2005), 1225-1243.  doi: 10.1090/S0002-9947-04-03543-3.

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E. N. DancerK. Geba and S. Rybicki, Classification of homotopy classes of equivariant gradient map, Fund. Math., 185 (2005), 1-18.  doi: 10.4064/fm185-1-1.

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E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, Ⅱ, Comm. Partial Differential Equations, 30 (2005), 1331-1358.  doi: 10.1080/03605300500258865.

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E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, J. Differential Equations, 210 (2005), 317-351.  doi: 10.1016/j.jde.2004.07.017.

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E. N. Dancer, Finite Morse index solutions of exponential problems, Ann. Inst. H. Poincare Anal. Non Linaire, 25 (2008), 173-179.  doi: 10.1016/j.anihpc.2006.12.001.

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E. N. Dancer, Finite Morse index solutions of supercritical problems, J. Reine Angew. Math., 620 (2008), 213-233.  doi: 10.1515/CRELLE.2008.055.

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E. N. Dancer and S. Yan, The Lazer-McKenna conjecture and a free boundary problem in two dimensions, J. Lond. Math. Soc., (2) 78 (2008), 639-662. doi: 10.1112/jlms/jdn045.

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E. N. Dancer and A. Farina, On the classification of solutions of $u = e^u$ on $R^N$: stability outside a compact set and applications, Proc. Amer. Math. Soc., 137 (2009), 1333-1338.  doi: 10.1090/S0002-9939-08-09772-4.

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E. N. DancerY. Du and Z. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations, 250 (2011), 3281-3310.  doi: 10.1016/j.jde.2011.02.005.

[52]

E. N. DancerK. Wang and Z. Zhang, Uniform Holder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, J. Differential Equations, 251 (2011), 2737-2769.  doi: 10.1016/j.jde.2011.06.015.

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E. N. DancerK. Wang and Z. Zhang, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture, J. Funct. Anal., 262 (2012), 1087-1131.  doi: 10.1016/j.jfa.2012.10.009.

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E. N. DancerK. Wang and Z. Zhang, Dynamics of strongly competing systems with many species, Tran. Amer. Math. Soc., 364 (2012), 961-1005.  doi: 10.1090/S0002-9947-2011-05488-7.

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E. N. Dancer, On the converse problem for the Gross-Pitaevskii equations with a large parameter, Discrete Contin. Dyn. Syst., 34 (2014), 2481-2493.  doi: 10.3934/dcds.2014.34.2481.

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D. Daners, Domain perturbation for linear and nonlinear parabolic equations, J. Differential Equations, 129 (1996), 358-402.  doi: 10.1006/jdeq.1996.0122.

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J. DávilaL. DupaigneK. Wang and J. Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285.  doi: 10.1016/j.aim.2014.02.034.

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C. Li and S. Li, The Fucik spectrum of Schrodinger operator and the existence of four solutions of Schrodinger equations with jumping nonlinearities, J. Differential Equations, 263 (2017), 7000-7097.  doi: 10.1016/j.jde.2017.07.038.

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G. LiS. Yan and J. Yang, The superlinear Lazer-McKenna conjecture for an elliptic problem with critical growth, Part Ⅱ, J. Differential Equations, 227 (2006), 301-332.  doi: 10.1016/j.jde.2006.02.011.

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G. LiS. Yan and J. Yang, The superlinear Lazer-McKenna conjecture for an elliptic problem with critical growth, Calc. Var. Partial Differential Equations, 28 (2007), 471-508.  doi: 10.1007/s00526-006-0051-z.

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[81]

B. NorisH. TavaresS. Terracini and G. Verzini, Uniform Hölder bounds for non-linear Schrödinger systems with strong competition, Commun. Pure Appl. Math., 63 (2010), 267-302.  doi: 10.1002/cpa.20309.

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[83]

K. PereraM. Squassina and Y. Yang, A note on the Dancer-Fucik spectra of the fractional p-Laplacian and Laplacian operators, Adv. Nonlinear Anal., 4 (2015), 13-23.  doi: 10.1515/anona-2014-0038.

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M. Schechter, The Fucik spectrum, Indiana Univ. Math. J., 43 (1994), 1139-1157.  doi: 10.1512/iumj.1994.43.43050.

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J. Wei and S. Yan, The Lazer-McKenna conjecture: the critical case, J. Funct. Anal., 244 (2007), 639-667.  doi: 10.1016/j.jfa.2006.11.002.

show all references

References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.

[2]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl.(4), 93 (1972), 231-246. doi: 10.1007/BF02412022.

[3]

L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $\mathbb R^3$ and a conjecture of De Giorgi, J. Am. Math. Soc., 13 (2000), 725-739.  doi: 10.1090/S0894-0347-00-00345-3.

[4]

A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294.  doi: 10.1002/cpa.3160410302.

[5]

Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, Springfield, MO, 2006.

[6]

K. Borsuk, Drei Sätze über die $n$-dimensionale Sphäre, Fund. Math., 20 (1933), 236-243.

[7]

B. BuffoniE. N. Dancer and J. F. Toland, The regularity and local bifurcation of steady periodic water waves, Arch. Ration. Mech. Anal., 152 (2000), 207-240.  doi: 10.1007/s002050000086.

[8]

B. BuffoniE. N. Dancer and J. F. Toland, The sub-harmonic bifurcation of Stokes waves, Arch. Ration. Mech. Anal., 152 (2000), 241-271.  doi: 10.1007/s002050000087.

[9]

D. Clark, A variant of the Lyusternik-Schnirelmann theory, Indiana Univ. Math. J., 22 (1973), 65-74.  doi: 10.1512/iumj.1973.22.22008.

[10]

C. Conley, Isolated Invariant Sets and The Morse Index, CBMS regional conference series in mathematics No. 38., Providence: Amer. Math. Soc., 1978.

[11]

M. ContiS. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560.  doi: 10.1016/j.aim.2004.08.006.

[12]

E. C. M. Crooks, E. N. Dancer and D. Hilhorst, On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions, Topol. Methods Nonlinear Anal., 30 (2007), 1-36.

[13]

M. CuestaD. de Figueiredo and J.-P. Gossez, The beginning of the Fucik Spectrum for the p-Laplacian, J. Differ. Equ., 159 (1999), 212-238.  doi: 10.1006/jdeq.1999.3645.

[14]

E. N. Dancer, Bifurcation theory in real Banach space, Proc. London Math. Soc., (3) 23 (1971), 699-734. doi: 10.1112/plms/s3-23.4.699.

[15]

E. N. Dancer, Bifurcation theory for analytic operators, Proc. London Math. Soc., (3) 26 (1973), 359-384. doi: 10.1112/plms/s3-26.2.359.

[16]

E. N. Dancer, Global structure of the solutions of non-linear real analytic eigenvalue problems, Proc. London Math. Soc., (3) 27 (1973), 747-765. doi: 10.1112/plms/s3-27.4.747.

[17]

E. N. Dancer, On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 76 (1977), 283-300.  doi: 10.1017/S0308210500019648.

[18]

E. N. Dancer, On the existence of solutions of certain asymptotically homogeneous problems, Math. Z., 177 (1981), 33-48.  doi: 10.1007/BF01214337.

[19]

E. N. Dancer, Symmetries, degree, homotopy indices and asymptotically homogeneous problems, Nonlinear Anal., 6 (1982), 667-686.  doi: 10.1016/0362-546X(82)90037-2.

[20]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.  doi: 10.1016/0022-247X(83)90098-7.

[21]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.  doi: 10.1090/S0002-9947-1984-0743741-4.

[22]

E. N. Dancer, On positive solutions of some pairs of differential equations, II, J. Differential Equations, 60 (1985), 236-258.  doi: 10.1016/0022-0396(85)90115-9.

[23]

E. N. Dancer, A new degree for $S^1$-invariant gradient mappings and applications, Ann. Inst. H. Poincare Anal. Non Linaire, 2 (1985), 329-370.  doi: 10.1016/S0294-1449(16)30396-1.

[24]

E. N. Dancer, Counterexamples to some conjectures on the number of solutions of nonlinear equations, Math. Ann., 272 (1985), 421-440.  doi: 10.1007/BF01455568.

[25]

E. N. Dancer, Multiple fixed points of positive mappings, J. Reine Angew. Math., 371 (1986), 46-66.  doi: 10.1515/crll.1986.371.46.

[26]

E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations, 74 (1988), 120-156.  doi: 10.1016/0022-0396(88)90021-6.

[27]

E. N. Dancer, On the influence of domain shape on the existence of large solutions of some superlinear problems, Math. Ann., 285 (1989), 647-669.  doi: 10.1007/BF01452052.

[28]

E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, Ⅱ, J. Differential Equations, 87 (1990), 316-339.  doi: 10.1016/0022-0396(90)90005-A.

[29]

E. N. Dancer and J. F. Toland, Degree theory for orbits of prescribed period of flows with the first integral, Proc. London Math. Soc., 60 (1990), 549-580.  doi: 10.1112/plms/s3-60.3.549.

[30]

E. N. Dancer and P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems, J. Reine Angew. Math., 419 (1991), 125-139.

[31]

E. N. Dancer, Generic domain dependence for nonsmooth equations and the open set problem for jumping nonlinearities, Topol. Methods Nonlinear Anal., 1 (1993), 139-150.  doi: 10.12775/TMNA.1993.011.

[32]

E. N. Dancer and Y. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations, 114 (1994), 434-475.  doi: 10.1006/jdeq.1994.1156.

[33]

E. N. Dancer and Y. Du, Existence of changing sign solutions for some semilinear problems with jumping nonlinearities at zero, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 1165-1176.  doi: 10.1017/S0308210500030171.

[34]

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