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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.

MSC:
 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
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