Chonghao Deng | Searcharxiv

arXiv2025

Normalized solutions to subcritical Choquard systems with double couplings

We consider the Choquard system with both linear and nonlinear couplings $-\Delta u + \mu_1 u =\lambda_1 ( I_\alpha * |u|^{r_1} ) |u|^{r_1-2} u + \beta p( I_\alpha * |v|^q)|u|^{p-2} u + \kappa v,$ $-\Delta v + \mu_2 v =\lambda_2 ( I_\alpha * |v|^{r_2} ) |v|^{r_2-2} v + \beta q( I_\alpha * |u|^p)|v|^{q-2} v + \kappa u , $ $\int_{\mathbb{R}^N} u^2 = \rho_1^2\, , \int_{\mathbb{R}^N} v^2 = \rho_2^2,$ where $N \in \{3,4\}$, $\lambda_1, \lambda_2, \beta, \kappa, \rho_1,\rho_2 > 0$, $2_{\alpha,*} :=\frac{N+\alpha}{N} <p,q , r_1, r_2 <2_\alpha^*:=\frac{N+\alpha}{N-2}$ and $p+q\leq 2r_1 \leq 2r_2$ . We investigate a classification result as the parameters $p+q$, $2r_1$ and $2r_2$ vary across the ranges $(\frac{2N+2\alpha}{N},\frac{2N+2\alpha+4}{N})$, $\{\frac{2N+2\alpha+4}{N}\}$, and $(\frac{2N+2\alpha+4}{N},\frac{2N+2\alpha}{N-2})$. Employing variational methods, we demonstrate the existence of a normalized ground state for the system in the mass subcritical, critical, and supercritical cases.

math.AP↗

arXiv2025

Normalized solutions to critical Choquard systems with linear and nonlinear couplings

We consider the critical Choquard system with both linear and nonlinear couplings $-\Delta v_1 + \mu_1 v_1 = ( I_\omega * |v_1|^{2_\omega^*} ) |v_1|^{2_\omega^* -2} v_1 + \theta p( I_\omega * |v_2|^q)|v_1|^{p-2} v_1 + \varepsilon v_2, \quad in \,\, \mathbb{R}^N, -\Delta v_2 + \mu_2 v_2 = ( I_\omega * |v_2|^{2_\omega^*} ) |v_2|^{2_\omega^* -2} v_2 + \theta q( I_\omega * |v_1|^p)|v_2|^{q-2} v_2 + \varepsilon v_1 , \quad in \,\, \mathbb{R}^N , \int_{\mathbb{R}^N} v_1^2 = \alpha_1^2\, , \int_{\mathbb{R}^N} v_2^2 = \alpha_2^2,$ where $N=3\,\, \text{or} \,\, 4$, $\alpha_1,\alpha_2 > 0 $, $\theta > 0 $, $2_{\omega,*} :=\frac{N+\omega}{N} 0$, $0<\omega \frac{2N+2\omega+4}{N}$, we apply variational methods to establish the existence of a positive normalized ground state for the system as $\theta>\theta_*,\;0<\varepsilon<\overline{\varepsilon}$.

math.AP↗