Multiple positive normalized solutions for nonlinear Schrödinger systems.

*(English)*Zbl 1396.35009Authors’ 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.

Reviewer: Patrick Winkert (Berlin)

##### MSC:

35J47 | Second-order elliptic systems |

35J60 | Nonlinear elliptic equations |

35J10 | Schrödinger operator, Schrödinger equation |

35J50 | Variational methods for elliptic systems |

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\textit{T. Gou} and \textit{L. Jeanjean}, Nonlinearity 31, No. 5, 2319--2345 (2018; Zbl 1396.35009)

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