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Normalized solutions for a system of coupled cubic Schrödinger equations on \(\mathbb R^3\). (English. French summary) Zbl 1347.35107

The authors are concerned with the system \[ \begin{aligned} -\Delta u-\lambda_1 u=\mu_1 u^3+\beta uv^2&\quad \text{in }\mathbb R^3,\\ -\Delta v-\lambda_2 v=\mu_2 v^3+\beta u^2v &\quad \text{in }\mathbb R^3,\end{aligned} \] such that \(\| u\|_{L^2(\mathbb R^3)}=a_1\), \(\| v\|_{L^2(\mathbb R^3)}=a_2\). When \(a_1\), \(a_2\), \(\lambda_1\), \(\lambda_2>0\) are fixed, is is established the existence of a solution \((u,v)\) to the above system for \(\beta>0\) in a certain range. These results are then extended to the case of systems with arbitrary number of components. Orbital stability for the corresponding standing waves is also discussed.

MSC:

35J50 Variational methods for elliptic systems
35J15 Second-order elliptic equations
35J60 Nonlinear elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
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[1] Ambrosetti, A.; Colorado, E., Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc. (2), 75, 1, 67-82, (2007) · Zbl 1130.34014
[2] Bartsch, T., Bifurcation in a multicomponent system of nonlinear Schrödinger equations, J. Fixed Point Theory Appl., 13, 1, 37-50, (2013) · Zbl 1281.35004
[3] Bartsch, T.; Dancer, N.; Wang, Z.-Q., A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differ. Equ., 37, 3-4, 345-361, (2010) · Zbl 1189.35074
[4] Bartsch, T.; De Valeriola, S., Normalized solutions of nonlinear Schrödinger equations, Arch. Math., 100, 1, 75-83, (2013) · Zbl 1260.35098
[5] Bartsch, T.; Jeanjean, L., Normalized solutions for nonlinear Schrödinger systems, (2015), Preprint
[6] Bartsch, T.; Wang, Z.-Q., Note on ground states of nonlinear Schrödinger systems, J. Partial Differ. Equ., 19, 3, 200-207, (2006) · Zbl 1104.35048
[7] Bartsch, T.; Wang, Z.-Q.; Wei, J., Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2, 353-367, (2007) · Zbl 1153.35390
[8] Bellazzini, J.; Jeanjean, L., On dipolar quantum gases in the unstable regime, (2014), Preprint
[9] Bellazzini, J.; Jeanjean, L.; Luo, T-J., Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations, Proc. Lond. Math. Soc., 107, 3, 303-339, (2013) · Zbl 1284.35391
[10] Berestycki, H.; Cazenave, T., Instabilité des états stationnaires dans LES équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math., 293, 9, 489-492, (1981) · Zbl 0492.35010
[11] Cazenave, T., Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, (2003), New York University, Courant Institute of Mathematical Sciences/American Mathematical Society New York/Providence, RI · Zbl 1055.35003
[12] Chen, Z.; Zou, W., An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differ. Equ., 48, 3-4, 695-711, (2013) · Zbl 1286.35104
[13] Correia, S., Stability of ground states for a system of m coupled semilinear Schrödinger equations, (2015), Preprint
[14] Esry, B. D.; Greene, C. H.; Burke, J. P.; Bohn, J. L., Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78, 3594, (1997)
[15] Ghoussoub, N., Duality and perturbation methods in critical point theory, Cambridge Tracts in Mathematics, vol. 107, (1993), Cambridge University Press Cambridge, with appendices by David Robinson · Zbl 0790.58002
[16] Ikoma, N., Uniqueness of positive solutions for a nonlinear elliptic system, NoDEA Nonlinear Differ. Equ. Appl., 16, 5, 555-567, (2009) · Zbl 1181.35063
[17] Ikoma, N., Compactness of minimizing sequences in nonlinear Schrödinger systems under multiconstraint conditions, Adv. Nonlinear Stud., 14, 1, 115-136, (2014) · Zbl 1297.35218
[18] Jeanjean, L., Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28, 10, 1633-1659, (1997) · Zbl 0877.35091
[19] Jeanjean, L.; Luo, T.-J.; Wang, Z.-Q., Multiple normalized solutions for quasi-linear Schrödinger equations, J. Differ. Equ., 259, 8, 3894-3928, (2015) · Zbl 1377.35074
[20] Kwong, M. K., Uniqueness of positive solutions of \(\operatorname{\Delta} u - u + u^p = 0\) in \(\mathbf{R}^n\), Arch. Ration. Mech. Anal., 105, 3, 243-266, (1989) · Zbl 0676.35032
[21] Le Coz, S., A note on berestycki-Cazenave’s classical instability result for nonlinear Schrödinger equations, Adv. Nonlinear Stud., 8, 3, 455-463, (2008) · Zbl 1156.35092
[22] Lin, T.-C.; Wei, J., Ground state of N coupled nonlinear Schrödinger equations in \(\mathbb{R}^n\), \(n \leq 3\), Commun. Math. Phys., 255, 3, 629-653, (2005) · Zbl 1119.35087
[23] Lin, T.-C.; Wei, J., Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 22, 4, 403-439, (2005) · Zbl 1080.35143
[24] Lions, P. L., The concentration-compactness principle in the calculus of variation. the locally compact case, part I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 1, 2, 109-145, (1984) · Zbl 0541.49009
[25] Lions, P. L., The concentration-compactness principle in the calculus of variation. the locally compact case, part II, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 1, 4, 223-283, (1984) · Zbl 0704.49004
[26] Liu, Z.; Wang, Z.-Q., Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10, 1, 175-193, (2010) · Zbl 1198.35067
[27] Luo, T.-J., Multiplicity of normalized solutions for a class of nonlinear Schrödinger-Poisson-Slater equations, J. Math. Anal. Appl., 416, 1, 195-204, (2014) · Zbl 1301.35155
[28] Maia, L. A.; Montefusco, E.; Pellacci, B., Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differ. Equ., 229, 2, 743-767, (2006) · Zbl 1104.35053
[29] Maia, L.d. A.; Montefusco, E.; Pellacci, B., Orbital stability property for coupled nonlinear Schrödinger equations, Adv. Nonlinear Stud., 10, 3, 681-705, (2010) · Zbl 1217.35175
[30] Malomed, B., Multi-component Bose-Einstein condensates: theory, (Kevrekidis, P. G.; Frantzeskakis, D. J.; Carretero-Gonzalez, R., Emergent Nonlinear Phenomena in Bose-Einstein Condensation, (2008), Springer-Verlag Berlin), 287-305 · Zbl 1151.82369
[31] Mandel, R., Minimal energy solutions for cooperative nonlinear Schrödinger systems, NoDEA Nonlinear Differ. Equ. Appl., 22, 2, 239-262, (2015) · Zbl 1312.35082
[32] Nguyen, N. V.; Wang, Z.-Q., Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system, Discrete Contin. Dyn. Syst., 36, 2, 1005-1021, (2016) · Zbl 1330.35411
[33] Nguyen, N. V.; Wang, Z.-Q., Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Differ. Equ., 16, 9-10, 977-1000, (2011) · Zbl 1252.35253
[34] Nguyen, N. V.; Wang, Z.-Q., Orbital stability of solitary waves of a 3-coupled nonlinear Schrödinger system, Nonlinear Anal., 90, 1-26, (2013) · Zbl 1281.35080
[35] Noris, B.; Tavares, H.; Terracini, S.; Verzini, G., Convergence of minimax structures and continuation of critical points for singularly perturbed systems, J. Eur. Math. Soc., 14, 4, 1245-1273, (2012) · Zbl 1248.35197
[36] Noris, B.; Tavares, H.; Verzini, G., Existence and orbital stability of the ground states with prescribed mass for the \(L^2\)-critical and supercritical NLS on bounded domains, Anal. PDE, 7, 8, 1807-1838, (2014) · Zbl 1314.35168
[37] Noris, B.; Tavares, H.; Verzini, G., Stable solitary waves with prescribed \(L^2\)-mass for the cubic Schrödinger system with trapping potentials, Discrete Contin. Dyn. Syst., Ser. A, 35, 12, 6085-6112, (2015) · Zbl 1336.35321
[38] Ohta, M., Stability of solitary waves for coupled nonlinear Schrödinger equations, Nonlinear Anal., 26, 5, 933-939, (1996) · Zbl 0860.35011
[39] Sato, Y.; Wang, Z.-Q., Least energy solutions for nonlinear Schrödinger systems with mixed attractive and repulsive couplings, Adv. Nonlinear Stud., 15, 1, 1-22, (2015) · Zbl 1316.35269
[40] Sirakov, B., Least energy solitary waves for a system of nonlinear Schrödinger equations in \(\mathbb{R}^n\), Commun. Math. Phys., 271, 1, 199-221, (2007) · Zbl 1147.35098
[41] Soave, N., On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition, Calc. Var. Partial Differ. Equ., 53, 3-4, 689-718, (2015) · Zbl 1323.35166
[42] Soave, N.; Tavares, H., New existence and symmetry results for least energy positive solutions of Schrödinger systems with mixed competition and cooperation terms, J. Differ. Equ., (2016) · Zbl 1337.35049
[43] Tavares, H.; Terracini, S., Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 29, 2, 279-300, (2012) · Zbl 1241.35046
[44] Terracini, S.; Verzini, G., Multipulse phases in k-mixtures of Bose-Einstein condensates, Arch. Ration. Mech. Anal., 194, 3, 717-741, (2009) · Zbl 1181.35069
[45] Wei, J.; Weth, T., Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Ration. Mech. Anal., 190, 1, 83-106, (2008) · Zbl 1161.35051
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