zbMATH — the first resource for mathematics

Existence and orbital stability of standing waves to nonlinear Schrödinger system with partial confinement. (English) Zbl 1436.35288
In this paper, the author studies the existence of weak solutions to the following semilinear Schrödinger system in \(\mathbb{R}^3\) with partially trapping potential \[ \left\{\begin{array}{ccc} -\Delta u_1 + (x_1^2+x_2^2)u_1 & = & \lambda_1 u_1 +\mu_1 |u_1|^{p_1-2}u_1 + \beta r_1|u_1|^{r_1-2}u_1|u_2|^{r_2}, \\-\Delta u_2 + (x_1^2+x_2^2)u_2 & = & \lambda_2 u_2 +\mu_2|u_2|^{p_2-2}u_2 + \beta r_2|u_1|^{r_1}|u_2|^{r_2-2}u_2, \end{array}\tag{1} \right. \] together with the \(L^2\)-normalization conditions \(\int_{\mathbb{R}^3}|u_1|^2\ dx=a_1\) and \(\int_{\mathbb{R}^3}|u_2|^2\ dx=a_2\) for given constants \(a_1, a_2>0\).
Such solutions give rise to standing-wave solutions \(\varphi_1(t,x)=e^{-i\lambda_1t}u_1(x)\), \(\varphi_2(t,x)=e^{-i\lambda_2t}u_2(x)\) of the corresponding time-dependent Schrödinger system: \[ \left\{ \begin{array}{ccc} -i\partial_t\varphi_1 + (x_1^2+x_2^2)\varphi_1 & = & \Delta \varphi_1 +\mu_1|\varphi_1|^{p_1-2}\varphi_1 + \beta r_1|\varphi_1|^{r_1-2}\varphi_1|\varphi_2|^{r_2}, \\-i\partial_t\varphi_2 + (x_1^2+x_2^2)\varphi_2 & = & \Delta \varphi_2 +\mu_2|\varphi_2|^{p_2-2}\varphi_2 + \beta r_2|\varphi_1|^{r_1}|\varphi_2|^{r_2-2}\varphi_2. \end{array} \right. \] The system (\(1\)) admits a variational formulation through the associated energy functional \[ J(u_1,u_2)=\frac12\int_{\mathbb{R}^3}|\nabla u_1|^2+|\nabla u_2|^2+(x_1^2+x_2^2)(|u_1|^2+|u_2|^2)dx-\frac{\mu_1}{p_1}\int_{\mathbb{R}^3}|u_1|^{p_1}dx-\frac{\mu_2}{p_2}\int_{\mathbb{R}^3}|u_2|^{p_2}dx-\beta\int_{\mathbb{R}^3}|u_1|^{r_1}|u_2|^{r_2}dx, \] defined on the energy space \(H=\{u\in H^1(\mathbb{R}^3)\,:\,\int_{\mathbb{R}^3}(x_1^2+x_2^2)|u|^2dx<\infty\}\).
The main result of the paper shows that, under the assumptions \(\mu_1, \mu_2,\beta >0\), \(2 < p_1\), \(p_2<\frac{10}3\), \(r_1, r_2>1\) and \(r_1+r_2<\frac{10}3\), the compactness up to \(x_3\)-translations of minimizing sequences of \(J\) restricted to the product \(S(a_1,a_2)=S(a_1)\times S(a_2)\) of \(L^2\)-spheres of radius \(a_1\) and \(a_2\) respectively.
In particular, the system (\(1\)), together with the normalization condition, has a solution \((u_1,u_2)\) which is a minimizer of \(J\), among all solutions satisfying the normalization condition.
Contrary to the case of a trapping potential \(V(x)=x_1^2+x_2^2+x_3^2\), the Sobolev embedding \(H\hookrightarrow L^2(\mathbb{R}^3)\) is not compact. In order to show the compactness of minimizing sequences of the restriction of \(J\), the author uses the Lions concentration-compactness principle [P.-L. Lions, Ann. Inst. H. Poincaré, Anal. Non Linéaire 1, 109–145 (1984; Zbl 0541.49009; ibid. 1, 223–283 (1984; Zbl 0704.49004)] and the more delicate part is to exclude dichotomy. For this step, the author makes use of a variant of the Brezis-Lieb lemma [H. Brézis and E. H. Lieb, Proc. Am. Math. Soc. 88, 486–490 (1983; Zbl 0526.46037)] as well as an argument based on the maximum principle and on the coupled rearrangement, a tool applied by M. Shibata [Math. Z. 287, No. 1–2, 341–359 (2017; Zbl 1382.35012)] to \(L^2\)-constrained problems, and which is based on the Steiner symmetrization. A key features of this rearrangement is the strict subadditivity \[ \int_{\mathbb{R}^3}|\nabla(u\ast v)|^2\ dx <\int_{\mathbb{R}^3}|u|^2\ dx+\int_{\mathbb{R}^3}|v|^2\ dx, \] for all positive functions \(u,v\in H^1(\mathbb{R}^3)\times C^1(\mathbb{R}^3)\) that are nonincreasing in the \(x_3\)-direction.
As a consequence of this main result, the author obtains in addition that minimizers of (1) are orbitally stable.

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q41 Time-dependent Schrödinger equations and Dirac equations
35J47 Second-order elliptic systems
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B35 Stability in context of PDEs
35J50 Variational methods for elliptic systems
35G50 Systems of nonlinear higher-order PDEs
35D30 Weak solutions to PDEs
35B50 Maximum principles in context of PDEs
Full Text: DOI
[1] Akhmediev, N.; Ankiewicz, A., Partially coherent solitons on a finite background, Phys. Rev. Lett., 82, 2661, (1999)
[2] Albert, J.; Bhattarai, S., Existence and stability of a two-parameter family of solitary waves for an NLS-KdV system, Adv. Differ. Equations, 18, 1129-1164, (2013) · Zbl 1290.35219
[3] Antonelli, P.; Carles, R.; Silva, J., Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Commun. Math. Phys., 334, 367-396, (2015) · Zbl 1309.35124
[4] Bellazzini, J.; Siciliano, G., Stable standing waves for a class of nonlinear Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62, 2, 267-280, (2011) · Zbl 1339.35280
[5] Bellazzini, J.; Boussaid, N.; Jeanjean, L.; Visciglia, N., Existence and stability of standing waves for supercritical NLS with a partial confinement, Commun. Math. Phys., 353, 229-251, (2017) · Zbl 1367.35150
[6] Benci, V.; Visciglia, N., Solitary waves with nonvanishing angular momentum, Adv. Nonlinear Stud., 3, 1, 151-160, (2003) · Zbl 1030.35051
[7] Bhattarai, S., Stability of solitary-wave solutions of coupled NLS equations with power-type nonlinearities, Adv. Nonlinear Anal., 4, 73-90, (2015) · Zbl 1315.35197
[8] Bhattarai, S., Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 36, 4, 1789-1811, (2016) · Zbl 1326.35331
[9] Brezis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., 88, 486-490, (1983) · Zbl 0526.46037
[10] Byeon, J., Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems, J. Differ. Equations, 163, 429-474, (2000) · Zbl 0952.35054
[11] Cazenave, T.; Lions, P.-L., Orbital stability of standing waves for some nonlinear Schröodinger equations, Commun. Math. Phys., 85, 549-561, (1982) · Zbl 0513.35007
[12] Chen, Z.; Zou, W., An optimal constant for the existence of least energy solutions of a coupled Schröinger system, Calculus Var. Partial Differ. Equations, 48, 3-4, 695-711, (2013) · Zbl 1286.35104
[13] Colin, M.; Jeanjean, L.; Squassina, M., Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23, 1353-1385, (2010) · Zbl 1192.35163
[14] Esry, B. D.; Greene, C. H.; Burke, J. P. Jr.; Bohn, J. L., Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78, 3594-3597, (1997)
[15] Frantzeskakis, D. J., Dark solitons in atomic Bose Einstein condensates: From theory to experiments, J. Phys. A: Math. Theor., 43, 213001, (2010) · Zbl 1192.82033
[16] Garrisi, D., On the orbital stability of standing-wave solutions to a coupled non-linear Klein-Gordon equation, Adv. Nonlinear Stud., 12, 639-658, (2012) · Zbl 1269.35039
[17] Gou, T.; Jeanjean, L., Existence and orbital stability of standing waves for nonlinear Schrödinger systems, Nonlinear Anal., 144, 10-22, (2016) · Zbl 06618019
[18] Ikoma, N., Compactness of minimizing sequences in nonlinear Schrödinger systems under multiconstraint conditions, Adv. Nonlinear Stud., 14, 115-136, (2014) · Zbl 1297.35218
[19] Lieb, E. H.; Loss, M., Analysis, (2001), American Mathematical Society: American Mathematical Society, Providence · Zbl 0966.26002
[20] Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. Part I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 109-145, (1984) · Zbl 0541.49009
[21] Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. Part II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 223-283, (1984) · Zbl 0704.49004
[22] Malomed, B.; Kevrekidis, P. G.; Frantzeskakis, D. J.; Carretero-Gonzalez, R., Multi-Component Bose-Einstein Condensates: Theory, 287-305, (2008), Springer-Verlag: Springer-Verlag, Berlin · Zbl 1151.82369
[23] Nguyen, N. V.; Wang, Z.-Q., Orbital stability of solitary waves for a nonlinear Schröinger system, Adv. Differ. Equations, 16, 977-1000, (2011) · Zbl 1252.35253
[24] 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
[25] Nguyen, N. V.; Wang, Z.-Q., Existence and stability of a two-parameter family of solitary waves for a 2-couple nonlinear Schrödinger system, Discrete Contin. Dyn. Syst., 36, 1005-1021, (2016) · Zbl 1330.35411
[26] Ohta, M., Stability of solitary waves for coupled nonlinear Schrödinger equations, Nonlinear Anal.: Theory, Methods Appl., 26, 933-939, (1996) · Zbl 0860.35011
[27] Ohta, M., “Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement,” e-print .
[28] Shibata, M., Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term, Manuscripta Math., 143, 221-237, (2014) · Zbl 1290.35252
[29] Shibata, M., A new rearrangement inequality and its application for L2-constraint minimizing problems, Math. Z., 287, 341-359, (2017) · Zbl 1382.35012
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.