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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References:

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