June  2014, 34(6): 2481-2493. doi: 10.3934/dcds.2014.34.2481

On the converse problem for the Gross-Pitaevskii equations with a large parameter

1. 

University of Sydeny, NSW 2006, Australia

Received  January 2013 Revised  April 2013 Published  December 2013

We show how certain solutions of the limit equation continue to solutions of the full equations when a parameter is large.
Citation: Norman E. Dancer. On the converse problem for the Gross-Pitaevskii equations with a large parameter. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2481-2493. doi: 10.3934/dcds.2014.34.2481
References:
[1]

M. D'Aujourd'hui, Sur L'ensemble de Résonance d'un Problème Demi-linéaire,, preprint, (1986). Google Scholar

[2]

T. Bartsch, N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, Calc. Var. Partial Differential Equations, 37 (2010), 345. doi: 10.1007/s00526-009-0265-y. Google Scholar

[3]

T. Bartsch and E. N. Dancer, Poincaré-Hopf type formulas on convex sets of Banach spaces,, Topological Methods in Nonlinear Analysis, 34 (2009), 213. Google Scholar

[4]

L. A. Caffarelli and A. Friedman, Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations,, J. Differential Equations, 60 (1985), 420. doi: 10.1016/0022-0396(85)90133-0. Google Scholar

[5]

H. Cartan, Calcul Différentiel,, Hermann, (1967). Google Scholar

[6]

E. N. Dancer, Stable and finite Morse index solutions on $\mathbbR^n$ or on bounded domains with small diffusion. II,, Indiana University Mathematics Journal, 53 (2004), 97. doi: 10.1512/iumj.2004.53.2354. Google Scholar

[7]

E. N. Dancer, On the indices of fixed points in cones and applications,, Journal of Mathematical Analysis, 91 (1983), 131. doi: 10.1016/0022-247X(83)90098-7. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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. doi: 10.1017/S0308210500030171. Google Scholar

[12]

E. N. Dancer and Z. M. Guo, Some remarks on the stability of sign changing solutions,, Tohoku Math. J. (2), 47 (1995), 199. doi: 10.2748/tmj/1178225592. Google Scholar

[13]

E. N. Dancer, K. Wang and Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species,, Journal of Differential Equations, 251 (2011), 2737. doi: 10.1016/j.jde.2011.06.015. Google Scholar

[14]

E. N. Dancer, K. Wang and Z. Zhang, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture,, Journal of Functional Analysis, 262 (2012), 1087. doi: 10.1016/j.jfa.2011.10.013. Google Scholar

[15]

E. N. Dancer and S. Yan, On the profile of the changing sign mountain pass solutions for an elliptic problem,, Trans. Amer. Math. Soc., 354 (2002), 3573. doi: 10.1090/S0002-9947-02-03026-X. Google Scholar

[16]

D. G. de Figueiredo and J.-P. Gossez, On the first curve of the Fučik spectrum of an elliptic operator,, Differential Integral Equations, 7 (1994), 1285. Google Scholar

[17]

T. Gallouët and O. Kavian, Résultats d'existence et de non-existence pour certains problèmes demi-linéaires à l'infini,, Ann. Fac. Sci. Toulouse Math. (5), 3 (1981), 201. doi: 10.5802/afst.568. Google Scholar

[18]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1981). Google Scholar

[19]

H. Hofer, A geometric description of the neighbourhood of a critical point given by the mountain pass theorem,, Journal London Mathematical Society (2), 31 (1985), 566. doi: 10.1112/jlms/s2-31.3.566. Google Scholar

[20]

R. Kohn and P. Sternberg, Local minimizers and singular perturbations,, Proc. Royal Society Edinburgh Sect. A, 111 (1989), 69. doi: 10.1017/S0308210500025026. Google Scholar

[21]

B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition,, Comm. Pure Applied Math., 63 (2010), 267. Google Scholar

[22]

B. Noris, H. Tavares, S. Terracini and G. Verzini, Convergence of minimax structures and continuation of critical points for singularly perturbed systems,, J. Eur. Math. Soc. (JEMS), 14 (2012), 1245. doi: 10.4171/JEMS/332. Google Scholar

[23]

R. Nussbaum, Some generalization of the Borsuk-Ulam theorem,, Proc. London Mathematical Society (3), 35 (1977), 136. Google Scholar

[24]

B. Ruf, On nonlinear elliptic problems with jumping nonlinearities,, Ann. Mat. Pura Appl. (4), 128 (1981), 133. doi: 10.1007/BF01789470. Google Scholar

[25]

G. Sweers, A sign-changing global minimizer on a convex domain,, in Progress in Partial Differential Equations: Elliptic and Parabolic Problems (Pont-à-Mousson, (1991), 251. Google Scholar

show all references

References:
[1]

M. D'Aujourd'hui, Sur L'ensemble de Résonance d'un Problème Demi-linéaire,, preprint, (1986). Google Scholar

[2]

T. Bartsch, N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, Calc. Var. Partial Differential Equations, 37 (2010), 345. doi: 10.1007/s00526-009-0265-y. Google Scholar

[3]

T. Bartsch and E. N. Dancer, Poincaré-Hopf type formulas on convex sets of Banach spaces,, Topological Methods in Nonlinear Analysis, 34 (2009), 213. Google Scholar

[4]

L. A. Caffarelli and A. Friedman, Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations,, J. Differential Equations, 60 (1985), 420. doi: 10.1016/0022-0396(85)90133-0. Google Scholar

[5]

H. Cartan, Calcul Différentiel,, Hermann, (1967). Google Scholar

[6]

E. N. Dancer, Stable and finite Morse index solutions on $\mathbbR^n$ or on bounded domains with small diffusion. II,, Indiana University Mathematics Journal, 53 (2004), 97. doi: 10.1512/iumj.2004.53.2354. Google Scholar

[7]

E. N. Dancer, On the indices of fixed points in cones and applications,, Journal of Mathematical Analysis, 91 (1983), 131. doi: 10.1016/0022-247X(83)90098-7. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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. doi: 10.1017/S0308210500030171. Google Scholar

[12]

E. N. Dancer and Z. M. Guo, Some remarks on the stability of sign changing solutions,, Tohoku Math. J. (2), 47 (1995), 199. doi: 10.2748/tmj/1178225592. Google Scholar

[13]

E. N. Dancer, K. Wang and Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species,, Journal of Differential Equations, 251 (2011), 2737. doi: 10.1016/j.jde.2011.06.015. Google Scholar

[14]

E. N. Dancer, K. Wang and Z. Zhang, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture,, Journal of Functional Analysis, 262 (2012), 1087. doi: 10.1016/j.jfa.2011.10.013. Google Scholar

[15]

E. N. Dancer and S. Yan, On the profile of the changing sign mountain pass solutions for an elliptic problem,, Trans. Amer. Math. Soc., 354 (2002), 3573. doi: 10.1090/S0002-9947-02-03026-X. Google Scholar

[16]

D. G. de Figueiredo and J.-P. Gossez, On the first curve of the Fučik spectrum of an elliptic operator,, Differential Integral Equations, 7 (1994), 1285. Google Scholar

[17]

T. Gallouët and O. Kavian, Résultats d'existence et de non-existence pour certains problèmes demi-linéaires à l'infini,, Ann. Fac. Sci. Toulouse Math. (5), 3 (1981), 201. doi: 10.5802/afst.568. Google Scholar

[18]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1981). Google Scholar

[19]

H. Hofer, A geometric description of the neighbourhood of a critical point given by the mountain pass theorem,, Journal London Mathematical Society (2), 31 (1985), 566. doi: 10.1112/jlms/s2-31.3.566. Google Scholar

[20]

R. Kohn and P. Sternberg, Local minimizers and singular perturbations,, Proc. Royal Society Edinburgh Sect. A, 111 (1989), 69. doi: 10.1017/S0308210500025026. Google Scholar

[21]

B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition,, Comm. Pure Applied Math., 63 (2010), 267. Google Scholar

[22]

B. Noris, H. Tavares, S. Terracini and G. Verzini, Convergence of minimax structures and continuation of critical points for singularly perturbed systems,, J. Eur. Math. Soc. (JEMS), 14 (2012), 1245. doi: 10.4171/JEMS/332. Google Scholar

[23]

R. Nussbaum, Some generalization of the Borsuk-Ulam theorem,, Proc. London Mathematical Society (3), 35 (1977), 136. Google Scholar

[24]

B. Ruf, On nonlinear elliptic problems with jumping nonlinearities,, Ann. Mat. Pura Appl. (4), 128 (1981), 133. doi: 10.1007/BF01789470. Google Scholar

[25]

G. Sweers, A sign-changing global minimizer on a convex domain,, in Progress in Partial Differential Equations: Elliptic and Parabolic Problems (Pont-à-Mousson, (1991), 251. Google Scholar

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