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Rushun Tian

Publications and source records attributed to Rushun Tian.

3 recordsLinked to original sources
arXiv2026

Ground state of indefinite coupled nonlinear Schr\"odinger systems

In this paper, we study the ground state solutions of the following coupled nonlinear Schr\"odinger system (P) $-\Delta u_1-\tau_1 u_1 =\mu_1u_1^3+\beta u_1u_2^2$, $ -\Delta u_2-\tau_2 u_2 =\mu_2u_2^3+\beta u_1^2u_2$ in $\Omega$, $u_1=u_2=0$ on $\partial\Omega$, where $\mu_1, \mu_2>0$, $\beta>0$ and $\Omega\subset \mathbb{R}^N (N\le3)$ is a bounded domain with smooth boundary. We are concerned with the indefinite case, i.e., $\tau_1, \tau_2$ are greater than or equal to the principal eigenvalue of $-\Delta$ with the Dirichlet boundary datum. By delicate variational arguments, we obtain the existence of ground state solution to $(P)$, and also provide information on critical energy levels for coupling parameter $\beta$ in some ranges.

math.AP↗
arXiv2015

Existence and bifurcation of solutions for a double coupled system of Schrodinger equations

Consider the following system of double coupled Schrödinger equations arising from Bose-Einstein condensates etc., \begin{equation*} \left\{\begin{array}{l} -Δu + u =μ_1 u^3 + βuv^2- κv, -Δv + v =μ_2 v^3 + βu^2v- κu, u\neq0, v\neq0\ \hbox{and}\ u, v\in H^1(\R^N), \end{array} \right. \end{equation*}where $μ_1, μ_2$ are positive and fixed, $κ$ and $β$ are linear and nonlinear coupling parameters respectively. We first use critical point theory and Liouville type theorem to prove some existence and nonexistence results on the positive solutions of this system. Then using the positive and non-degenerate solution of the scalar equation $-Δω+ω=ω^3$, $ω\in H_r^1(\R^N)$, we construct a synchronized solution branch to prove that for $β$ in certain range and fixed, there exist a series of bifurcations in product space $\R\times H^1_r(\R^N)\times H^1_r(\R^N)$ with parameter $κ$.

math.AP↗
arXiv2014

Bifurcations for a Coupled Schrödinger System with Multiple Components

In this paper, we study local bifurcations of an indefinite elliptic system with multiple components: \begin{equation*} \left\{\begin{array}{ll} -Δu_j + au_j = μ_ju_j^3+β\sum_{k\ne j}u_k^2u_j, u_j>0\ \ \hbox{in}\ Ω, u_j=0 \ \ \hbox{on}\ \partialΩ,\ j=1,\dots,n. \end{array} \right. \end{equation*} Here $Ω\subset{\mathbb{R}}^N$ is a smooth and bounded domain, $n\ge3$, $a<-Λ_1$ where $Λ_1$ is the principal eigenvalue of $(-Δ, H_0^1(Ω))$; $μ_j$ and $β$ are real constants. Using the positive and non-degenerate solution of the scalar equation $-Δω-ω=-ω^3$, $ω\in H_0^1(Ω)$, we construct a synchronized solution branch $\mathcal{T}_ω$. Then we find a sequence of local bifurcations with respect to $\mathcal{T}_ω$, and we find global bifurcation branches of partially synchronized solutions.

math.AP↗