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Juncheng Wei

Publications and source records attributed to Juncheng Wei.

283 records · Page 16Linked to original sources
arXiv2010

Interface Foliation Near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature

Let $(\MM ,{\tilde g})$ be an $N$-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation $$ ε^2Δ_{ {\tilde g}} {u}\,+\, (1 - {u}^2)u \,=\,0\quad \mbox{in } \MM, $$ where $ε$ is a small parameter. Let $\KK\subset \MM$ be an $(N-1)$-dimensional smooth minimal submanifold that separates $\MM$ into two disjoint components. Assume that $\KK$ is non-degenerate in the sense that it does not support non-trivial Jacobi fields, and that $|A_{\KK}|^2+\mbox{Ric}_{\tilde g}(ν_{\KK}, ν_{\KK})$ is positive along $\KK$. Then for each integer $m\geq 2$, we establish the existence of a sequence $ε= ε_j\to 0$, and solutions $u_ε$ with $m$-transition layers near $\KK$, with mutual distance $O(ε|\ln ε|)$.

math.AP↗
arXiv2009

Homoclinic solutions for fourth order traveling wave equations

We consider homoclinic solutions of fourth order equations $$ u^{""} + β^2 u^{"} + V_u (u)=0 {in} \R ,$$ where $V(u)$ is either the suspension bridge type $V(u)=e^u-1-u$ or Swift-Hohenberg type $ V(u)= {1/4}(u^2-1)^2$. For the suspension bridge type equation, we prove existence of a homoclinic solution for {\em all} $ β\in (0, β_*)$ where $ β_{*}= 0.7427...$. For the Swift-Hohenberg type equation, we prove existence of a homoclinic solution for each $β\in (0, β_{*})$, where $β_{*}=0.9342...$. This partially solves a conjecture of Chen--McKenna \cite{YCM1}.

math.AP↗
arXiv2009

On De Giorgi Conjecture in Dimension $N \geq 9$

A celebrated conjecture due to De Giorgi states that any bounded solution of the equation $Δu + (1-u^2) u = 0 \hbox{in} \R^N $ with $\pp_{y_N}u >0$ must be such that its level sets $\{u=\la\}$ are all hyperplanes, {\em \bf at least} for dimension $N\le 8$. A counterexample for $N\ge 9$ has long been believed to exist. Based on a minimal graph $Γ$ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in $\R^N$, $N\ge 9$, we prove that for any small $α>0$ there is a bounded solution $u_α(y)$ with $\pp_{y_N}u_α>0$, which resembles $ \tanh (\frac t{\sqrt{2}}) $, where $t=t(y)$ denotes a choice of signed distance to the blown-up minimal graph $Γ_α:= α^{-1}Γ$. This solution constitutes a counterexample to De Giorgi conjecture for $N\ge 9$.

math.AP↗
arXiv2009

Entire Solutions of the Allen-Cahn equation and Complete Embedded Minimal Surfaces of Finite Total Curvature in $\R^3$

We consider minimal surfaces $M$ which are complete, embedded and have finite total curvature in $\R^3$, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation $Δu + f(u) = 0 \hbox{in} \R^3 $. Here $f=-W'$ with $W$ bistable and balanced, for instance $W(u) =\frac 14 (1-u^2)^2$. We assume that $M$ has $m\ge 2$ ends, and additionally that $M$ is non-degenerate, in the sense that its bounded Jacobi fields are all originated from rigid motions (this is known for instance for a Catenoid and for the Costa-Hoffman-Meeks surface of any genus). We prove that for any small $α>0$, the Allen-Cahn equation has a family of bounded solutions depending on $m-1$ parameters distinct from rigid motions, whose level sets are embedded surfaces lying close to the blown-up surface $M_α:= α^{-1} M$, with ends possibly diverging logarithmically from $M_\A$. We prove that these solutions are $L^\infty$-{\em non-degenerate} up to rigid motions, and find that their Morse index coincides with the index of the minimal surface. Our construction suggests parallels of De Giorgi conjecture for general bounded solutions of finite Morse index.

math.AP↗
arXiv2008

Infinitely many solution for prescribed curvature problem on $S^N$

We consider the following prescribed scalar curvature problem on $ S^N$ (*)$$\left\{\begin{array}{l} - Δ_{S^N} u + \frac{N(N-2)}{2} u = \tilde{K} u^{\frac{N+2}{N-2}} {on} S^N, u >0 \end{array}\right. $$ where $ \tilde{K}$ is positive and rotationally symmetric. We show that if $\tilde{K}$ has a local maximum point between the poles then equation (*) has {\bf infinitely many non-radial positive} solutions, whose energy can be made arbitrarily large.

math.AP↗
arXiv2008

Infinitely many positive solutions for the nonlinear Shcrodinger equations in $R^N$

We consider the following nonlinear problem in $\R^N$ $$\label{eq} - Δu +V(|y|)u=u^{p},\quad u>0 {in} \R^N, u \in H^1(\R^N) $$ where $V(r)$ is a positive function, $1 0$, $m>1$, $θ>0$, and $V_0>0$, such that \[ V(r)= V_0+\frac a {r^m} +O\bigl(\frac1{r^{m+θ}}\bigr),\quad \text{as $r\to +\infty$,} \] then \eqref{eq} has {\bf infinitely many non-radial positive} solutions, whose energy can be made arbitrarily large.

math.AP↗
arXiv2007

The Toda system and multiple-end solutions of autonomous planar elliptic problems

We construct a new class of positive solutions for a classical semilinear elliptic problem in the plane which arise for instance as the standing-wave problem for the standard nonlinear Schrödinger equation or in nonlinear models in Turing's theory biological theory of pattern formation such as the Gray-Scott or Gierer-Meinhardt systems. The solutions we construct have the property that their energy over a ball of radius R grows linearly with R as R tends to infinity. These solutions are strongly related to the solutions of a Toda system.

math.AP↗
arXiv2007

Properties of positive solutions of an Elliptic Equation with negative exponents

In this paper, we study the existence and non-existence result of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics. We can get some useful local gradient estimate and L1 lower bound for positive solutions of the elliptic equation. A uniform positive lower bound for convex positive solutions is also obtained. We show that in lower dimensions, there is no stable positive solutions in the whole space. In the whole space of dimension two, we can show that there is no positive smooth solution with finite Morse index. Symmetry properties of related integral equations are also given.

math.AP↗
arXiv2007

Half-Skyrmions and Spike-Vortex Solutions of Two-Component Nonlinear Schrodinger Systems

Recently, skyrmions with integer topological charges have been observed numerically but have not yet been shown rigorously on two-component systems of nonlinear Schrodinger equations (NLSE) describing a binary mixture of Bose-Einstein condensates. Besides, half-skyrmions characterized by half-integer topological charges can also be found in the nonlinear sigma model which is a model of the Bose-Einstein condensate of the Schwinger bosons. Here we prove rigorously the existence of half-skyrmions which may come from a new type of soliton solutions called spike-vortex solutions of two-component systems of NLSE on the entire plane. These spike-vortex solutions having spikes in one component and a vortex in the other component may form half-skyrmions. By Liapunov-Schmidt reduction process, we may find spike-vortex solutions of two-component systems of NLSE.

math.AP↗
arXiv2007

Transition Layer for the Heterogeneous Allen-Cahn Equation

We consider the equation $\e^{2}Δu=(u-a(x))(u^2-1)$ in $Ω$, $\frac{\partial u}{\partial ν} =0$ on $\partial Ω$, where $Ω$ is a smooth and bounded domain in $\R^n$, $ν$ the outer unit normal to $\paΩ$, and $a$ a smooth function satisfying $-1 0} and {a<0}. Assuming $\nabla a \neq 0$ on $K$ and $a\ne 0$ on $\partial Ω$, we show that there exists a sequence $\e_j \to 0$ such that the above equation has a solution $u_{\e_j}$ which converges uniformly to $\pm 1$ on the compact sets of $Ø_{\pm}$ as $j \to + \infty$.

math.AP↗
arXiv2006

Symbiotic Bright Solitary Wave Solutions of Coupled Nonlinear Schrodinger Equations

Conventionally, bright solitary wave solutions can be obtained in self-focusing nonlinear Schrodinger equations with attractive self-interaction. However, when self-interaction becomes repulsive, it seems impossible to have bright solitary wave solution. Here we show that there exists symbiotic bright solitary wave solution of coupled nonlinear Schrodinger equations with repulsive self-interaction but strongly attractive interspecies interaction. For such coupled nonlinear Schrodinger equations in two and three dimensional domains, we prove the existence of least energy solutions and study the location and configuration of symbiotic bright solitons. We use Nehari's manifold to construct least energy solutions and derive their asymptotic behaviors by some techniques of singular perturbation problems.

math.AP↗
arXiv2005

Mesa-type patterns in the one-dimensional Brusselator and their stability

The Brusselator is a generic reaction-diffusion model for a tri-molecular chemical reaction. We consider the case when the input and output reactions are slow. In this limit, we show the existence of $K$-periodic, spatially bi-stable structures, \emph{mesas}, and study their stability. Using singular perturbation techniques, we find a threshold for the stability of $K$ mesas. This threshold occurs in the regime where the exponentially small tails of the localized structures start to interact. By comparing our results with Turing analysis, we show that in the generic case, a Turing instability is followed by a slow coarsening process whereby logarithmically many mesas are annihilated before the system reaches a steady equilibrium state. We also study a ``breather''-type instability of a mesa, which occurs due to a Hopf bifurcation. Full numerical simulations are shown to confirm the analytical results.

nlin.PS↗