arXiv subjects

Duokui Yan

Publications and source records attributed to Duokui Yan.

16 recordsLinked to original sources

arXiv2025

Positive normalized solutions of Schr\"{o}dinger equations with Sobolev critical growth in bounded domains

This paper investigates the existence of positive normalized solutions to the Sobolev critical Schr\"{o}dinger equation: \begin{equation*} \left\{ \begin{aligned} &-\Delta u +\lambda u =|u|^{2^*-2}u \quad &\mbox{in}& \ \Omega,\\ &\int_{\Omega}|u|^{2}dx=c, \quad u=0 \quad &\mbox{on}& \ \partial\Omega, \end{aligned} \right. \end{equation*} where $\Omega\subset\mathbb{R}^{N}$ ($N\geq3$) is a bounded smooth domain, $2^*=\frac{2N}{N-2}$, $\lambda\in \mathbb{R}$ is a Lagrange multiplier, and $c>0$ is a prescribed constant. By introducing a novel blow-up analysis for Sobolev subcritical approximation solutions with uniformly bounded Morse index and fixed mass, we establish the existence of mountain pass type positive normalized solutions for $N\ge 3$. This resolves an open problem posed in [Pierotti, Verzini and Yu, SIAM J. Math. Anal. 2025].

math.AP↗

arXiv2023

Convergence of least energy sign-changing solutions for logarithmic Schr\"{o}dinger equations on locally finite graphs

In this paper, we study the following logarithmic Schr\"{o}dinger equation \[ -\Delta u+\lambda a(x)u=u\log u^2\ \ \ \ \mbox{ in }V \] on a connected locally finite graph $G=(V,E)$, where $\Delta$ denotes the graph Laplacian, $\lambda > 0$ is a constant, and $a(x) \geq 0$ represents the potential. Using variational techniques in combination with the Nehari manifold method based on directional derivative, we can prove that, there exists a constant $\lambda_0>0$ such that for all $\lambda\geq\lambda_0$, the above problem admits a least energy sign-changing solution $u_{\lambda}$. Moreover, as $\lambda\to+\infty$, we prove that the solution $u_{\lambda}$ converges to a least energy sign-changing solution of the following Dirichlet problem \[\begin{cases} -\Delta u=u\log u^2~~~&\mbox{ in }\Omega,\\ u(x)=0~~~&\mbox{ on }\partial\Omega, \end{cases}\] where $\Omega=\{x\in V: a(x)=0\}$ is the potential well.

math.AP↗

arXiv2022

Ground states for logarithmic Schr\"{o}dinger equations on locally finite graphs

In this paper, we study the following logarithmic Schr\"{o}dinger equation \[ -\Delta u+a(x)u=u\log u^2\ \ \ \ \mbox{in }V, \] where $\Delta$ is the graph Laplacian, $G=(V,E)$ is a connected locally finite graph, the potential $a: V\to \mathbb{R}$ is bounded from below and may change sign. We first establish two Sobolev compact embedding theorems in the case when different assumptions are imposed on $a(x)$. It leads to two kinds of associated energy functionals, one of which is not well-defined under the logarithmic nonlinearity, while the other is $C^1$. The existence of ground state solutions are then obtained by using the Nehari manifold method and the mountain pass theorem respectively.

math.AP↗

arXiv2022

Normalized ground state solutions of nonlinear Schr\"odinger equations involving exponential critical growth

We are concerned with the following nonlinear Schr\"odinger equation \begin{eqnarray*} \begin{aligned} \begin{cases} -\Delta u+\lambda u=f(u) \ \ {\rm in}\ \mathbb{R}^{2},\\ u\in H^{1}(\mathbb{R}^{2}),~~~ \int_{\mathbb{R}^2}u^2dx=\rho, \end{cases} \end{aligned} \end{eqnarray*} where $\rho>0$ is given, $\lambda\in\mathbb{R}$ arises as a Lagrange multiplier and $f$ satisfies an exponential critical growth. Without assuming the Ambrosetti-Rabinowitz condition, we show the existence of normalized ground state solutions for any $\rho>0$. The proof is based on a constrained minimization method and the Trudinger-Moser inequality in $\mathbb{R}^2$.

math.AP↗

arXiv2021

Existence of hyperbolic motions to a class of Hamiltonians and generalized $N$-body system via a geometric approach

For the classical $N$-body problem in $\mathbb{R}^d$ with $d\ge2$, Maderna-Venturelli in their remarkable paper [Ann. Math. 2020] proved the existence of hyperbolic motions with any positive energy constant, starting from any configuration and along any non-collision configuration. Their original proof relies on the long time behavior of solutions by Chazy 1922 and Marchal-Saari 1976, on the H\"{o}lder estimate for Ma\~{n}\'{e}'s potential by Maderna 2012, and on the weak KAM theory. We give a new and completely different proof for the above existence of hyperbolic motions. The central idea is that, via some geometric observation, we build up uniform estimates for Euclidean length and angle of geodesics of Ma\~{n}\'{e}'s potential starting from a given configuration and ending at the ray along a given non-collision configuration. Note that we do not need any of the above previous studies used in Maderna-Venturelli's proof. Moreover, our geometric approach works for Hamiltonians $\frac12\|p\|^2-F(x)$, where $F(x)\ge 0$ is lower semicontinuous and decreases very slowly to $0$ faraway from collisions. We therefore obtain the existence of hyperbolic motions to such Hamiltonians with any positive energy constant, starting from any admissible configuration and along any non-collision configuration. Consequently, for several important potentials $F\in C^{2}(\Omega)$, we get similar existence of hyperbolic motions to the generalized $N$-body system $\ddot{x} = \nabla_x F(x)$, which is an extension of Maderna-Venturelli [Ann. Math. 2020].

math.DS↗

arXiv2020

Geometric properties of minimizers in the planar three-body problem with two equal masses

It it shown that each lobe of the figure-eight orbit is star-shaped, which implies the polar angle is monotone in each lobe. In general, it is not clear when a minimizer is star-shaped. In this paper, we study minimizers connecting two fixed-ends (i.e. the Bolza problem) in the planar three-body problem with two equal masses. We show that if the Jacobi coordinates of the two fixed-ends are in adjacent closed quadrants, then the corresponding minimizer must stay in two adjacent closed quadrants. If we further assume the two Jacobi coordinates are orthogonal on one of the fixed-ends, then the polar angles of the Jacobi coordinates in the minimizer have at most one critical point. If the two Jacobi coordinates are orthogonal on both ends, then the two polar angles must be monotone. These geometric properties can be applied to show the existence of two sets of periodic orbits.

math.DS↗

arXiv2017

Existence of prograde double-double orbits in the equal-mass four-body problem

By introducing simple topological constraints and applying a binary decomposition method, we show the existence of a set of prograde double-double orbits for any rotation angle $θ\in (0, π/7]$ in the equal-mass four-body problem. A new geometric argument is introduced to show that for any $θ\in (0, π/2)$, the action of the minimizer corresponding to the prograde double-double orbit is strictly greater than the action of the minimizer corresponding to the retrograde double-double orbit. This geometric argument can also be applied to study orbits in the planar three-body problem, such as the retrograde orbits, the prograde orbits, the Schubart orbit and the Hénon orbit.

math.DS↗

arXiv2017

The Broucke-Hénon orbit and the Schubart Orbit in the planar three-body problem with equal masses

In this paper, we study the variational properties of two special orbits: the Schubart orbit and the Broucke-Hénon orbit. We show that under an appropriate topological constraint, the action minimizer must be either the Schubart orbit or the Broucke-Hénon orbit. One of the main challenges is to prove that the Schubart orbit coincides with the action minimizer connecting a collinear configuration with a binary collision and an isosceles configuration. A new geometric argument is introduced to overcome this challenge.

math.DS↗

arXiv2017

Action minimizers under topological constraints in the planar equal-mass four-body problem

It is shown that in the planar equal-mass four-body problem, there exist two sets of new action minimizers connecting two planar boundary configurations with fixed symmetry axes and specific order constraints on the four bodies: a double isosceles configuration and an isosceles trapezoid configuration. By applying the level estimate method, these minimizers are shown to be collision-free and they can be extended to two new sets of periodic or quasi-periodic orbits.

math.DS↗

arXiv2014

A type of multiple integral with loggamma function

In this paper, we study the multiple integral $ \displaystyle I= \int_0^1 \int_0^1 \dots \int_0^1 f(x_1+x_2 + \dots +x_n) \, dx_1 \, dx_2 \, \dots \, dx_n$. A general formula of $I$ is presented. As an application, the integral $I$ with $f(x)= \log Γ(x)$ is evaluated. We show that the values of $I$ share a common formula for all $n \in \mathbb{N}$. The subsidiary computational challenges are substantial and interesting in their own right.

math.CA↗

arXiv2014

New phenomenons in the spatial isosceles three-body problem

In this work, we study the periodic orbits in the spatial isosceles three-body problem. These periodic orbits form a one-parameter set with a rotation angle $θ$ as the parameter. Some new phenomenons are discovered by applying our numerical method. The periodic orbit coincides with the planar Euler orbit when $0 < θ\leq 0.32 π$ and it changes to a spatial orbit when $0.33 π\leq θ< π$. Eventually, the spatial orbit becomes a planar collision orbit when $θ=π$. Furthermore, an oscillated behavior is found when $θ=π/2$, which is chaotic but bounded under a small perturbation. As another application of our numerical method, 7 new periodic orbits are presented in the end.

nlin.CD↗

arXiv2011

Existence and Stability of Symmetric Periodic Simultaneous Binary Collision Orbits in the Planar Pairwise Symmetric Four-Body Problem

We extend our previous analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized fully symmetric equal mass four-body problem to the analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise symmetric equal mass four-body problem. We then use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar pairwise symmetric 1, m, 1, m four-body problem for $m$ between 0 and 1. Numerical estimates of the the characteristic multipliers show that these periodic orbits are linearly stability when $0.54\leq m\leq 1$, and are linearly unstable when $0<m\leq0.53$.

math.DS↗

arXiv2010

Simultaneous Binary Collisions for Collinear Four-Body Problem

In this paper, we use canonical transformations to collectively analytically continue the singularities of the simultaneous binary collision solutions for the collinear four- body problem in both the decoupled case and the coupled case. All the solutions are found and more importantly, we describe the relationship between the decoupled solutions and the coupled solutions.

math.DS↗

arXiv2010

Periodic Solutions with Singularities in Two Dimensions in the $n$-body Problem

Analytical methods are used to prove the existence of a periodic, symmetric solution with singularities in the planar 4-body problem. A numerical calculation and simulation are used to generate the orbit. The analytical method easily extends to any even number of bodies. Multiple simultaneous binary collisions are a key feature of the orbits generated.

math.DS↗

arXiv2009

Linear Stability for Some Symmetric Periodic Simultaneous Binary Collision Orbits in the Four-Body Problem

We apply the analytic-numerical method of Roberts to determine the linear stability of time-reversible periodic simultaneous binary collision orbits in the symmetric collinear four body problem with masses 1, m, m, 1, and also in a symmetric planar four-body problem with equal masses. For the collinear problem, this verifies the earlier numerical results of Sweatman for linear stability.

math.DS↗

arXiv2008

Periodic Solutions with Alternating Singularities in the Collinear Four-body Problem

This paper shows the existence of a periodic orbit with singularity in the symmetric collinear four body problem. In each period of the orbit, there is a binary collision (BC) between the inner two bodies and a simultaneous binary collision (SBC) of the two clusters on both sides of the origin. The system is regularized and the existence is proven by using the implicit function theorem and a continuity argument on differential equations of the regularized Hamiltonian.

math.DS↗