# zbMATH — the first resource for mathematics

Multiple positive normalized solutions for nonlinear Schrödinger systems. (English) Zbl 1396.35009
Authors’ abstract: We consider the existence of multiple positive solutions to the nonlinear Schrödinger systems set on $$H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)$$, $\begin{cases} -\Delta u_1 =\lambda_1 u_1+\mu_1 |u_1|^{p_1-2}u_1+\beta r_1|u_1|^{r_1-2}u_1|u_2|^{r_2},\\ -\Delta u_2 =\lambda_2 u_2+\mu_2 |u_2|^{p_2-2}u_2+\beta r_2|u_1|^{r_1}|u_2|^{r_2-2}u_2, \end{cases}$ under the constraint \begin{aligned} \int_{\mathbb{R}^N} |u_1|^2 dx=a_1, \quad \int_{\mathbb{R}^N} |u_2|^2 dx =a_2. \end{aligned} Here $$a_1, a_2>0$$ are prescribed, $$\mu_1,\mu_2,\beta>0$$, and the frequencies $$\lambda_1,\lambda_2$$ are unknown and will appear as Lagrange multipliers. Two cases are studied, the first when \begin{aligned} N\geq 1,\;2<p_1,p_2<2+\frac{4}{N},\;r_1,r_2>1, \;2+\frac{4}{N}< r_1+r_2< 2^*, \end{aligned} the second when \begin{aligned} N\geq 1, \;2+\frac{4}{N}<p_1, p_2 <2^*, \;r_1,r_2>1, \;r_1+r_2<2+\frac{4}{N}. \end{aligned} In both cases, assuming that $$\beta>0$$ is sufficiently small, we prove the existence of two positive solutions. The first one is a local minimizer for which we establish the compactness of the minimizing sequences and also discuss the orbital stability of the associated standing waves. The second solution is obtained through a constrained mountain pass and a constrained linking respectively.

##### MSC:
 35J47 Second-order elliptic systems 35J60 Nonlinear elliptic equations 35J10 Schrödinger operator, Schrödinger equation 35J50 Variational methods for elliptic systems
Full Text:
##### References:
 [1] Akhmediev N and Ankiewicz A 1999 Partially coherent solitons on a finite background Phys. Rev. Lett.82 2661 [2] Ambrosetti A and Colorado E 2007 Standing waves of some coupled nonlinear Schrödinger equations J. London Math. Soc.75 67-82 · Zbl 1130.34014 [3] Bartsch T and De Valeriola S 2013 Normalized solutions of nonlinear Schrödinger equations Arch. Math.100 75-83 · Zbl 1260.35098 [4] Bartsch T and Jeanjean L 2018 Normalized solutions for nonlinear Schrödinger systems Proc. R. Soc. Edinburgh 148 225-42 [5] Bartsch T, Jeanjean L and Soave N 2016 Normalized solutions for a system of coupled cubic Schrödinger equations on R3 J. Math. Pures. Appl.106 583-614 · Zbl 1347.35107 [6] Bartsch T and Soave N 2017 A natural constraint approach to normalized solutions on nonlinear Schrödinger equations and systems J. Funct. Anal.272 4998-5037 · Zbl 06714264 [7] Bartsch T and Soave N 2017 Multiple normalized solutions for a competing system of Schrödinger equations (arXiv:1703.02832) [8] Bhattarai S 2016 Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations Discrete Contin. Dyn. Syst. A 36 1789-11 · Zbl 1326.35331 [9] Bellazzini J and Jeanjean L 2016 On dipolar quantum gases in the unstable regime SIAM J. Math. Anal.48 2028-58 · Zbl 1352.35157 [10] Bellazzini J, Boussaid N, Jeanjean L and Visciglia N 2017 Existence and stability of standing waves for supercritical NLS with a partial confinement Commun. Math. Phys.353 229-51 · Zbl 1367.35150 [11] Bellazzini J, Jeanjean L and Luo T 2013 Existence and instability of standing waves with prescribed norm for a class of Schröinger-Poisson equations Proc. London Math. Soc.107 303-39 · Zbl 1284.35391 [12] Brézis H and Lieb E 1983 A relation between pointwise convergence of functions and convergence of functionals Proc. Am. Math. Soc.88 486-90 [13] Busca J and Sirakov B 2000 Symmetry results for semilinear elliptic systems in the whole space J. Differ. Equ.163 41-56 · Zbl 0952.35033 [14] Cao D, Chern I-L and Wei J 2011 On ground state of spinor Bose-Einstein condensates Nonlinear Differ. Equ. Appl.18 427-45 · Zbl 1228.35218 [15] Cazenave T and Lions P-L 1982 Orbital stability of standing waves for some nonlinear Schrödinger equations Commun. Math. Phys.85 549-61 [16] Esry B D, Greene C H, Burke J P Jr and Bohn J L 1997 Hartree-Fock theory for double condensates Phys. Rev. Lett.78 3594-7 [17] Frantzeskakis D J 2010 Dark solitons in atomic Bose-Einstein condensates: from theory to experiments J. Phys. A: Math. Theor.43 213001 · Zbl 1192.82033 [18] Ghoussoub N 1993 Duality and Perturbation Methods in Critical Point Theorey (Cambridge: Cambridge University Press) · Zbl 0790.58002 [19] Gou T and Jeanjean L 2016 Existence and orbital stability of standing waves for nonlinear Schrödinger systems Nonlinear Anal.144 10-22 · Zbl 06618019 [20] Han Q and Lin F 1997 Elliptic Partial Differetial Equations [21] Ikoma N 2014 Compactness of minimizing sequences in nonlinear Schrödinger systems under multiconstraint conditions Adv. Nonlinear Stud.14 115-36 · Zbl 1297.35218 [22] Jeanjean L 1997 Existence of solutions with prescribed norm for semilinear elliptic equations Nonlinear Anal.28 1633-59 · Zbl 0877.35091 [23] Jeanjean L and Squassina M 2011 An approach to minimization under a constraint: the added mass technique Calc. Var. PDE 41 511-34 · Zbl 1223.49021 [24] Lieb E H and Loss M 2001 Analysis(Graduate Studies in Mathematics vol 14) 2nd edn (Providence, RI: American Mathematical Society) [25] Lions P-L 1984 The concentration-compactness principle in the calculus of variations. The locally compact case, Part I Ann. Inst. Henri Poincare Anal. Non Linéaire 1 109-45 [26] Lions P-L 1984 The concentration-compactness principle in the calculus of variations. The locally compact case, Part II Ann. Inst. Henri Poincare Anal. Non Linéaire 1 223-83 [27] Lin T-C and Wei J 2005 Ground state of N coupled nonlinear Schrödinger equations in Rn,n⩽3 Commun. Math. Phys.255 629-53 [28] Noris B, Tavares H and Verzini G 2014 Existence and orbital stability of the ground states with prescribed mass for the L2 critical and supercritical NLS on bounded domains Anal. PDE 7 1807-38 · Zbl 1314.35168 [29] Noris B, Tavares H and Verzini G 2015 Stable solitary waves with prescribed L2-mass for the cubic Schrödinger system with trapping potentials Discrete Contin. Dyn. Syst. A 35 6085-112 · Zbl 1336.35321 [30] Maia L A, Montefusco E and Pellacci B 2006 Positive solutions for a weakly coupled nonlinear Schrödinger system J. Differ. Equ.229 743-67 · Zbl 1104.35053 [31] Nguyen N V, Tian R, Deconinck B and Sheils N 2013 Global existence of a coupled system of Schrödinger equations with power-type nonlinearities J. Math. Phys.54 011503 · Zbl 1286.35230 [32] Nguyen N V and Wang Z-Q 2016 Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system Discrete Contin. Dyn. Syst.36 1005-21 [33] Ohta M 1996 Stability of solitary waves for coupled nonlinear Schrödinger equations Nonlinear Anal.26 933-9 [34] Pierotti D and Verzini G 2017 Normalized bound states for the nonlinear Schrödinger equation in bounded domains Calc. Var. Part. Diff. Equ.56 · Zbl 1420.35374 [35] Shibata M 2016 A new rearrangement inequality and its application for L2-constraint minimizing problems Math. Z.287 341-59 [36] Sirakov B 2007 Least energy solitary waves for a system of nonlinear Schröodinger equations in Rn Commun. Math. Phys.271 199-221 · Zbl 1147.35098 [37] Timmermans E 1998 Phase separation of Bose-Einstein condensates Phys. Rev. Lett.81 5718-21
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.