American Institute of Mathematical Sciences

November  2019, 12(7): 1835-1839. doi: 10.3934/dcdss.2019120

On a degree associated with the Gross-Pitaevskii system with a large parameter

 School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia

Received  November 2017 Revised  July 2018 Published  December 2018

Fund Project: Partially supported by the Australian Research Council.

In a number of cases we calculate the sum of the degrees of the small positive solutions of the Gross-Pitaevskii system when the interaction is strong.

Citation: E. Norman Dancer. On a degree associated with the Gross-Pitaevskii system with a large parameter. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1835-1839. doi: 10.3934/dcdss.2019120
References:
 [1] E. N. Dancer, On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 76 (1976/77), 283-300.  doi: 10.1017/S0308210500019648.  Google Scholar [2] ______, 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.  Google Scholar [3] 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.  Google Scholar [4] E. N. Dancer, K. 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.2011.10.013.  Google Scholar [5] S. Fučík, Boundary value problems with jumping nonlinearities, Časopis Pěst. Mat., 101 (1976), 69-87.  Google Scholar [6] B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.  doi: 10.1002/cpa.20309.  Google Scholar [7] R. D. Nussbaum, Some generalizations of the Borsuk-Ulam theorem, Proc. London Math. Soc., (3), 35 (1977), 136-158. doi: 10.1112/plms/s3-35.1.136.  Google Scholar

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References:
 [1] E. N. Dancer, On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 76 (1976/77), 283-300.  doi: 10.1017/S0308210500019648.  Google Scholar [2] ______, 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.  Google Scholar [3] 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.  Google Scholar [4] E. N. Dancer, K. 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.2011.10.013.  Google Scholar [5] S. Fučík, Boundary value problems with jumping nonlinearities, Časopis Pěst. Mat., 101 (1976), 69-87.  Google Scholar [6] B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.  doi: 10.1002/cpa.20309.  Google Scholar [7] R. D. Nussbaum, Some generalizations of the Borsuk-Ulam theorem, Proc. London Math. Soc., (3), 35 (1977), 136-158. doi: 10.1112/plms/s3-35.1.136.  Google Scholar
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