Linlin Sun
In this paper we introduce a flow to study the Toda system, which we call {\it Toda flow.} More generally, we introduce a flow of the Liouville systems, formulated as a coupled parabolic system with nonlocal interactions. Finite-time…
We study degenerate quasilinear elliptic equations on Riemannian manifolds and obtain several Liouville theorems. Notably, we provide rigorous proof asserting the nonexistence of positive solutions to the subcritical Lane-Emden-Fowler…
Let $(M,g)$ be a compact Riemann surface with unit area. We investigate the mean field equation for equilibrium turbulence: \begin{align} \begin{cases} -\Delta u = \rho_1\left(\frac{h_1e^{u}}{\int_Mh_1e^udv_g}-1\right) -…
We study semilinear elliptic equations on finite graphs with fully general exponential nonlinearities, thereby extending classical equations such as the Kazdan-Warner and Chern-Simons equations. A key contribution of this work is the…
In this paper, we investigate critical quasilinear elliptic partial differential equations on a complete Riemannian manifold with nonnegative Ricci curvature. By exploiting a new and sharp nonlinear Kato inequality and establishing some…
In this paper, we focus on the sinh-Gordon equation on graphs. We introduce a uniform a priori estimate to define the topological degree for this equation with nonzero prescribed functions on finite, connected and symmetric graphs.…
For a homotopy class $[u]$ of maps between a closed Riemannian manifold $M$ and a general manifold $N$, we want to find a Dirac-harmonic map with the map component in the given homotopy class. Most known results require the index to be…
In this paper, we study the following Kazdan-Warner equation with sign-changing prescribed function $h$ \begin{align*} -\Delta u=8\pi\left(\frac{he^{u}}{\int_{\Sigma}he^{u}}-1\right) \end{align*} on a closed Riemann surface whose area is…
Let $(M, g)$ be a compact Riemann surface with area $1$. We investigate the Toda system \begin{align} \begin{cases} -\Delta u_1 = 2\rho_1(h_1e^{u_1}-1) - \rho_2(h_2e^{u_2}-1),\\ -\Delta u_2 = 2\rho_2(h_2e^{u_2}-1) - \rho_1(h_1e^{u_1}-1),…
The incorporation of ceramics into polymers, forming solid composite electrolytes (SCEs) leads to enhanced electrical performance of all-solid-state lithium metal batteries. This is because the dispersed ceramics particles increase the…
Let $(V,E)$ be a finite connected graph. We are concerned about the Chern-Simons Higgs model $$\Delta u=\lambda e^u(e^u-1)+f, \quad\quad\quad\quad\quad\quad{(0.1)}$$ where $\Delta$ is the graph Laplacian, $\lambda$ is a real number and $f$…
Let $(\Sigma,g)$ be a compact Riemann surface with smooth boundary $\partial\Sigma$, $\Delta_g$ be the Laplace-Beltrami operator, and $h$ be a positive smooth function. Using a min-max scheme introduced by Djadli-Malchiodi (2006) and Djadli…
As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. Recently, Guo \cite{Guo} proved…
We study Kazdan-Warner equations on a connected finite graph via the method of the degree theory. Firstly, we prove that all solutions to the Kazdan-Warner equation with nonzero prescribed function are uniformly bounded and the Brouwer…
We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface $(\Sigma,g)$ \begin{align*} -\Delta_{g}u=8\pi\left(\frac{he^{u}}{\int_{\Sigma}he^{u}{\rm d}\mu_{g}}-\frac{1}{\int_{\Sigma}{\rm…
Let $\Sigma$ be a closed Riemann surface, $h$ a positive smooth function on $\Sigma$, $\rho$ and $\alpha$ real numbers. In this paper, we study a generalized mean field equation \begin{align*} -\Delta u=\rho\left(\dfrac{he^u}{\int_\Sigma…
In this paper, we firstly prove that every hyper-Lagrangian submanifold $L^{2n} (n > 1)$ in a hyperk\"ahler $4n$-manifold is a complex Lagrangian submanifold. Secondly, we demonstrate an optimal rigidity theorem with the condition on the…
In this note, we study minimal Lagrangian surfaces in $\mathbb{B}^4$ with Legendrian capillary boundary on $\mathbb{S}^3$. On the one hand, we prove that any minimal Lagrangian surface in $\mathbb{B}^4$ with Legendrian free boundary on…
In this paper, we firstly establish a new volume growth estimate for spacelike entire graphs in the pseudo-Euclidean space $\mathbb{R}^{m+n}_n$. Then by using this volume growth estimate and the Co-Area formula, we prove various rigidity…
In the finite blocklength scenario, which is suitable for practical applications, a method of maximizing the average effective secrecy rate (AESR) is proposed for a UAV-enabled secure communication by optimizing the UAV's trajectory and…