# American Institute of Mathematical Sciences

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, 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, Ecole Polytechnique de Lausanne, 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), 345-361. 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), 213-229.  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), 420-433. doi: 10.1016/0022-0396(85)90133-0.  Google Scholar [5] H. Cartan, Calcul Différentiel, Hermann, Paris, 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), 97-108. 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), 131-151. 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), 120-156. 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), 236-258. 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), 434-475. 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), 1165-1176. 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), 199-225. 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), 2737-2769. 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), 1087-1131. 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), 3573-3600. 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), 1285-1302.  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), 201-246. doi: 10.5802/afst.568.  Google Scholar [18] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 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), 566-570. 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), 69-84. 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), 267-302.  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), 1245-1273. doi: 10.4171/JEMS/332.  Google Scholar [23] R. Nussbaum, Some generalization of the Borsuk-Ulam theorem, Proc. London Mathematical Society (3), 35 (1977), 136-158.  Google Scholar [24] B. Ruf, On nonlinear elliptic problems with jumping nonlinearities, Ann. Mat. Pura Appl. (4), 128 (1981), 133-151. 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), Pitman Research Notes Math. Ser., 266, Longman Sci. Tech., Harlow, 1992, 251-258.  Google Scholar

show all references

##### References:
 [1] M. D'Aujourd'hui, Sur L'ensemble de Résonance d'un Problème Demi-linéaire, preprint, Ecole Polytechnique de Lausanne, 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), 345-361. 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), 213-229.  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), 420-433. doi: 10.1016/0022-0396(85)90133-0.  Google Scholar [5] H. Cartan, Calcul Différentiel, Hermann, Paris, 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), 97-108. 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), 131-151. 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), 120-156. 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), 236-258. 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), 434-475. 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), 1165-1176. 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), 199-225. 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), 2737-2769. 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), 1087-1131. 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), 3573-3600. 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), 1285-1302.  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), 201-246. doi: 10.5802/afst.568.  Google Scholar [18] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 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), 566-570. 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), 69-84. 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), 267-302.  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), 1245-1273. doi: 10.4171/JEMS/332.  Google Scholar [23] R. Nussbaum, Some generalization of the Borsuk-Ulam theorem, Proc. London Mathematical Society (3), 35 (1977), 136-158.  Google Scholar [24] B. Ruf, On nonlinear elliptic problems with jumping nonlinearities, Ann. Mat. Pura Appl. (4), 128 (1981), 133-151. 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), Pitman Research Notes Math. Ser., 266, Longman Sci. Tech., Harlow, 1992, 251-258.  Google Scholar
 [1] Ruizhao Zi. Global solution in critical spaces to the compressible Oldroyd-B model with non-small coupling parameter. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6437-6470. doi: 10.3934/dcds.2017279 [2] Yonggeun Cho, Tohru Ozawa. On small amplitude solutions to the generalized Boussinesq equations. Discrete & Continuous Dynamical Systems, 2007, 17 (4) : 691-711. doi: 10.3934/dcds.2007.17.691 [3] Yang Cao, Jingxue Yin. Small perturbation of a semilinear pseudo-parabolic equation. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 631-642. doi: 10.3934/dcds.2016.36.631 [4] Marcello D'Abbicco. Small data solutions for semilinear wave equations with effective damping. Conference Publications, 2013, 2013 (special) : 183-191. doi: 10.3934/proc.2013.2013.183 [5] Ling-Jun Wang. The dynamics of small amplitude solutions of the Swift-Hohenberg equation on a large interval. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1129-1156. doi: 10.3934/cpaa.2012.11.1129 [6] Adriana Buică, Jean–Pierre Françoise, Jaume Llibre. Periodic solutions of nonlinear periodic differential systems with a small parameter. Communications on Pure & Applied Analysis, 2007, 6 (1) : 103-111. doi: 10.3934/cpaa.2007.6.103 [7] Deconinck Bernard, Olga Trichtchenko. High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1323-1358. doi: 10.3934/dcds.2017055 [8] Roger Grimshaw, Dmitry Pelinovsky. Global existence of small-norm solutions in the reduced Ostrovsky equation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 557-566. doi: 10.3934/dcds.2014.34.557 [9] Salomón Alarcón. Multiple solutions for a critical nonhomogeneous elliptic problem in domains with small holes. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1269-1289. doi: 10.3934/cpaa.2009.8.1269 [10] Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651 [11] Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217 [12] Daniele Bartoli, Leo Storme. On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric. Advances in Mathematics of Communications, 2014, 8 (3) : 271-280. doi: 10.3934/amc.2014.8.271 [13] Fioralba Cakoni, Shari Moskow, Scott Rome. The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions. Inverse Problems & Imaging, 2015, 9 (3) : 725-748. doi: 10.3934/ipi.2015.9.725 [14] Gengsheng Wang, Guojie Zheng. The optimal control to restore the periodic property of a linear evolution system with small perturbation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1621-1639. doi: 10.3934/dcdsb.2010.14.1621 [15] Uchida Hidetake. Analytic smoothing effect and global existence of small solutions for the elliptic-hyperbolic Davey-Stewartson system. Conference Publications, 2001, 2001 (Special) : 182-190. doi: 10.3934/proc.2001.2001.182 [16] Meina Gao. Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 173-204. doi: 10.3934/dcds.2015.35.173 [17] Yuji Sagawa, Hideaki Sunagawa. The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5743-5761. doi: 10.3934/dcds.2016052 [18] Michael Röckner, Rongchan Zhu, Xiangchan Zhu. A remark on global solutions to random 3D vorticity equations for small initial data. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4021-4030. doi: 10.3934/dcdsb.2019048 [19] Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407 [20] Mikhail Kamenskii, Boris Mikhaylenko. Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 437-452. doi: 10.3934/dcdsb.2013.18.437

2019 Impact Factor: 1.338