Zhaoyang Yun | Searcharxiv

arXiv2024

Normalized solutions to Schrödinger systems with potentials

In this paper, we study the normalized solutions of the Schrödinger system with trapping potentials \begin{equation}\label{eq:diricichlet} \begin{cases} -Δu_1+V_1(x)u_1-λ_1 u_1=μ_1 u_1^3+βu_1u_2^{2}+κu_2~\hbox{in}~ \mathbb{R}^3,\\ -Δu_2+V_2(x)u_2-λ_2 u_2=μ_2 u_2^3+βu_1^2u_2+κu_1~\hbox{in}~ \mathbb{R}^3, u_1\in H^1(\mathbb{R}^3), u_2\in H^1(\mathbb{R}^3),\nonumber \end{cases} \end{equation} under the constraint \begin{equation} \int_{\mathbb{R}^3} u_1^2=a_1^2,~\int_{\mathbb{R}^3} u_2^2=a_2^2\nonumber, \end{equation} where $μ_1,μ_2,a_1,a_2,β>0$, $κ\in\mathbb{R}$, $V_1(x)$ and $V_2(x)$ are trapping potentials, and $λ_1,λ_2$ are lagrangian multipliers, this is a typical $L^2$-supercritical case in $\mathbb{R}^3$. We obtain the existence of solutions to this system by minimax theory on the manifold for $κ=0$ and $κ\neq 0$ respectively.

math.AP↗

arXiv2021

Normalized solutions to Schrödinger systems with linear and nonlinear couplings

In this paper, we study important Schrödinger systems with linear and nonlinear couplings \begin{equation}\label{eq:diricichlet} \begin{cases} -Δu_1-λ_1 u_1=μ_1 |u_1|^{p_1-2}u_1+r_1β|u_1|^{r_1-2}u_1|u_2|^{r_2}+κ(x)u_2~\hbox{in}~\mathbb{R}^N,\\ -Δu_2-λ_2 u_2=μ_2 |u_2|^{p_2-2}u_2+r_2β|u_1|^{r_1}|u_2|^{r_2-2}u_2+κ(x)u_1~ \hbox{in}~\mathbb{R}^N,\\ u_1\in H^1(\mathbb{R}^N), u_2\in H^1(\mathbb{R}^N),\nonumber \end{cases} \end{equation} with the condition $$\int_{\mathbb{R}^N} u_1^2=a_1^2, \int_{\mathbb{R}^N} u_2^2=a_2^2,$$ where $N\geq 2$, $μ_1,μ_2,a_1,a_2>0$, $β\in\mathbb{R}$, $2<p_1,p_2<2^*$, $2<r_1+r_2<2^*$, $κ(x)\in L^{\infty}(\mathbb{R}^N)$ with fixed sign and $λ_1,λ_2$ are Lagrangian multipliers. We use Ekland variational principle to prove this system has a normalized radially symmetric solution for $L^2-$subcritical case when $N\geq 2$, and use minimax method to prove this system has a normalized radially symmetric positive solution for $L^2-$supercritical case when $N=3$, $p_1=p_2=4,\ r_1=r_2=2$.

math.AP↗