Existence of normalized solutions for the coupled elliptic system with quadratic nonlinearity. (English) Zbl 1487.35207
Summary: In the present paper, we study the existence of the normalized solutions for the following coupled elliptic system with quadratic nonlinearity
\[
\begin{cases}
-\Delta u-\lambda_1 u=\mu_1 |u| u+\beta uv & \text{in }\mathbb{R}^N, \\
-\Delta v-\lambda_2 v=\mu_2 |v| v+\frac{\beta}{2} u^2 & \text{in }\mathbb{R}^N,
\end{cases}
\]
where \(u,v\) satisfying the additional condition
\[
\int\limits_{\mathbb{R}^N} u^2 \mathrm{d}x = a_1,\quad \int\limits_{\mathbb{R}^N} v^2 \mathrm{d}x= a_2.
\]
On the one hand, we prove the existence of minimizer for the system with \(L^2\)-subcritical growth \((N\leq 3)\). On the other hand, we prove the existence results for different ranges of the coupling parameter \(\beta > 0\) with \(L^2\)-supercritical growth \((N=5)\). Our argument is based on the rearrangement techniques and the minimax construction.
MSC:
| 35J61 | Semilinear elliptic equations |
| 35J20 | Variational methods for second-order elliptic equations |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 49J40 | Variational inequalities |
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