arXiv subjects

Lun Guo

Publications and source records attributed to Lun Guo.

5 recordsLinked to original sources

arXiv2026

Qualitative analysis of energy ground states for magnetic focusing Gross-Pitaevskii equations

We prove the uniqueness, asymptotics, symmetry, and orbital stability of energy ground states for 3D magnetic Gross-Pitaevskii equations under mild conditions on the electric potential and magnetic field. In particular, both the electric potential and the magnetic field are allowed to have singularities, and we cover in a unified approach the physically relevant Aharonov-Bohm magnetic field as well as the constant magnetic field. Since we are in the 3D case, the unique energy ground state (up to a phase factor) is obtained as a local minimizer, rather than a global one, by restricting the kinetic energy of candidate critical points within a suitable range. The qualitative analysis of the energy ground state we carry on is mainly based on a related Pohozaev identity, the implicit function theorem, and variational methods.

math.AP↗

arXiv2026

Uniqueness and non-uniqueness of least energy normalized solutions for nonlinear Schr\"odinger equations on compact metric graphs

We investigate uniqueness and non-uniqueness of least energy normalized and least energy nodal normalized solutions for nonlinear Schr\"odinger equations on compact metric graphs. We first prove existence of least energy nodal normalized solutions in the $L^2$-subcritical regime and, at the critical exponent, below a graph-dependent threshold. We then establish a conditional non-uniqueness result for slightly $L^2$-subcritical powers and identify broad classes of graphs for which the required condition either holds or fails. In particular, we prove uniqueness of least energy nodal normalized solutions on the interval for every mass and every $p\in(2,6)$. Finally, using ODE and phase-plane techniques, we show that least energy normalized solutions on the interval are unique for $p$ sufficiently close to $2$. Overall, the results reveal a strong dependence of the uniqueness picture on the nonlinearity power, the topology, and the metric of the graph, and a structural difference between the constant sign and the sign-changing settings.

math.AP↗

arXiv2022

A coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent: existence and multiplicity of high energy positive solutions

This paper deals with a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent \begin{equation*} \begin{cases} -Δu+(V_1(x)+λ_1)u=μ_1(|x|^{-4}*u^{2})u+β(|x|^{-4}*v^{2})u, \ \ &x\in R^N, -Δv+(V_2(x)+λ_2)v=μ_2(|x|^{-4}*v^{2})v+β(|x|^{-4}*u^{2})v, \ \ &x\in R^N, \end{cases} \end{equation*} where $N\geq 5$, $λ_1$, $λ_2\geq 0$ with $λ_1+λ_2\neq 0$, $V_1(x), V_{2}(x)\in L^{\frac{N}{2}}(R^N)$ are nonnegative functions and $μ_1$, $μ_2$, $β$ are positive constants. Such system arises from mathematical models in Bose-Einstein condensates theory and nonlinear optics. By variational methods combined with degree theory, we prove some results about the existence and multiplicity of high energy positive solutions under the hypothesis $β>\max\{μ_1,μ_2\}$

math.AP↗

arXiv2022

Standing waves for two-component elliptic system with critical growth in $\mathbb{R}^{4}$: the attractive case

In this paper, we consider the following two-component elliptic system with critical growth \begin{equation*} \begin{cases} -Δu+(V_1(x)+λ)u=μ_1u^{3}+βuv^{2}, \ \ x\in \mathbb{R}^4, -Δv+(V_2(x)+λ)v=μ_2v^{3}+βvu^{2}, \ \ x\in \mathbb{R}^4 , % u\geq 0, \ \ v\geq 0 \ \text{in} \ \R^4. \end{cases} \end{equation*} where $V_j(x) \in L^{2}(\mathbb{R}^4)$ are nonnegative potentials and the nonlinear coefficients $β,μ_j$, $j=1,2$, are positive. Here we also assume $λ>0$. By variational methods combined with degree theory, we prove some results about the existence and multiplicity of positive solutions under the hypothesis $β>\max\{μ_1,μ_2\}$. These results generalize the results for semilinear Schrödinger equation on half space by Cerami and Passaseo (SIAM J. Math. Anal., 28, 867-885, (1997)) to the above elliptic system, while extending the existence result from Liu and Liu (Calc. Var. Partial Differential Equations, 59:145, (2020)).

math.AP↗

arXiv2017

Existence and asymptotic behavior of the least energy solutions for fractional Choquard equations with potential well

In this paper, we are concerned with the existence and asymptotic behavior of least energy solutions for following nonlinear Choquard equation driven by fractional Laplacian $$(-Δ)^{s} u+λV(x)u=(I_α\ast F(u))f(u) \ \ in \ \ R^{N},$$ where $N> 2s$, $ (N-4s)^{+}<α< N$, $λ$ is a positive parameter and the nonnegative potential function $V(x)$ is continuous. By variational methods, we prove the existence of least energy solution which localize near the potential well $int (V^{-1}(0))$ as $λ$ large enough.

math.AP↗