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Juncheng Wei

Publications and source records attributed to Juncheng Wei.

At least 271 records · Page 15Linked to original sources
arXiv2013

Serrin's Overdetermined Problem and Constant Mean Curvature Surfaces

For all $N \geq 9$, we find smooth entire epigraphs in $\R^N$, namely smooth domains of the form $Ω: = \{x\in \R^N\ / \ x_N > F (x_1,\ldots, x_{N-1})\}$, which are not half-spaces and in which a problem of the form $Δu + f(u) = 0 $ in $Ω$ has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on $\partial Ω$. This answers negatively for large dimensions a question by Berestycki, Caffarelli and Nirenberg \cite{bcn2}. In 1971, Serrin \cite{serrin} proved that a bounded domain where such an overdetermined problem is solvable must be a ball, in analogy to a famous result by Alexandrov that states that an embedded compact surface with constant mean curvature (CMC) in Euclidean space must be a sphere. In lower dimensions we succeed in providing examples for domains whose boundary is close to large dilations of a given CMC surface where Serrin's overdetermined problem is solvable.

math.AP↗
arXiv2013

Infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials

We consider the standing-wave problem for a nonlinear Schrödinger equation, corresponding to the semilinear elliptic problem \begin{equation*} -Δu+V(x)u=|u|^{p-1}u,\ u\in H^1(\mathbb{R}^2), \end{equation*} where $V(x)$ is a uniformly positive potential and $p>1$. Assuming that \begin{equation*} V(x)=V_\infty+\frac{a}{|x|^m}+O\Big(\frac{1}{|x|^{m+σ}}\Big),\ \text{as}\ |x|\rightarrow+\infty, %\tag{$V2$} \end{equation*} for instance if $p>2$, $m>2$ and $σ>1$ we prove the existence of infinitely many positive solutions. If $V(x)$ is radially symmetric, this result was proved in \cite{WY-10}. The proof without symmetries is much more difficult, and for that we develop a new {\em intermediate Lyapunov-Schmidt reduction method}, which is a compromise between the finite and infinite dimensional versions of it.

math.AP↗
arXiv2013

Concentrating standing waves for the fractional nonlinear Schrödinger equation

We consider the semilinear equation $$ ε^{2s} (-Δ)^s u + V(x)u - u^p = 0, \quad u>0, \quad u\in H^{2s}(\R^N) $$ where $0 0$, and $ε>0$ is a small number. Letting $w_λ$ be the radial ground state of $(-Δ)^s w_λ+ λw_λ- w_λ^p=0$ in $H^{2s}(\R^N)$, we build solutions of the form $$ u_ε(x) \sim \sum_{i=1}^k w_{λ_i} ((x-ξ_i^ε)/ε),$$ where $λ_i = V(ξ_i^ε)$ and the $ξ_i^ε$ approach suitable critical points of $V$. Via a Lyapunov Schmidt variational reduction, we recover various existence results already known for the case $s=1$. In particular such a solution exists around $k$ nondegenerate critical points of $V$. For $s=1$ this corresponds to the classical results by Floer-Weinstein and Oh.

math.AP↗
arXiv2013

Qualitative Analysis of Rupture Solutions for an MEMS Problem

We prove a sharp Hölder continuity estimates of rupture sets for sequences of solutions of the following nonlinear problem with negative exponent $$ Δu= \frac{1}{u^p}, \ p>1, \ \mbox{in} \ Ω.$$ As a consequence, we prove the existence of rupture solutions with isolated ruptures in a bounded convex domain in $\R^2$.

math.AP↗
arXiv2013

A Monotonicity Formula and a Liouville-type Theorem for a Fourth Order Supercritical Problem

We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ Δ^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$ where $ p>1$ and $n\ge1$. We give a complete classification of stable and finite Morse index solutions (whether positive or sign changing), in the full exponent range. We also compute an upper bound of the Hausdorff dimension of the singular set of extremal solutions. Our approach is motivated by Fleming's tangent cone analysis technique for minimal surfaces and Federer's dimension reduction principle in partial regularity theory. A key tool is the monotonicity formula for biharmonic equations.

math.AP↗
arXiv2013

Nonlocal Minimal Lawson Cones

We prove the existence of the analog of Lawson's minimal cones for a notion of nonlocal minimal surface introduced by Caffarelli, Roquejoffre and Savin, and establish their stability/instability in low dimensions. In particular we find that there are nonlocal stable minimal cones in dimension 7, in contrast with the case of classical minimal surfaces.

math.AP↗
arXiv2012

On solutions with polynomial growth to an autonomous nonlinear elliptic problem

We study the following nonlinear elliptic problem [-Δu =F^{'} (u) in {\mathbb R}^n] where $F(u)$ is a periodic function. Moser (1986) showed that for any minimal and nonself-intersecting solution, there exist $ α\in {\mathbb R}^n$ and $ C>0$ such that [(*) | u- α\cdot x | \leq C.] He also showed the existence of solutions with any prescribed $α\in {\mathbb R}^n$. In this note, we first prove that any solution satisfying (*) with nonzero vector $α$ must be one dimensional. Then we show that in ${\mathbb R}^2$, for any positive integer $d\geq 1$ there exists a solution with polynomial growth $|x|^d$.

math.AP↗
arXiv2012

Infinitely many positive solutions for nonlinear equations with non-symmetric potential

We consider the following nonlinear Schrodinger equation [{l} Δu-(1+δV)u+f(u)=0 in \R^N, u>0 in \R^N, u\in H^1(\R^N).] where $V$ is a potential satisfying some decay condition and $ f(u)$ is a superlinear nonlinearity satisfying some nondegeneracy condition. Using localized energy method, we prove that there exists some $δ_0$ such that for $0<δ<δ_0$, the above problem has infinitely many positive solutions. This generalizes and gives a new proof of the results by Cerami-Passaseo-Solimini (CPAM to appear). The new techniques allow us to establish the existence of infinitely many positive bound states for elliptic systems.

math.AP↗
arXiv2012

Regularization of point vortices for the Euler equation in dimension two, part II

In this paper, we continue to construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure now is carried out by constructing solutions to the following elliptic problem \[ {cases} -\ep^2 Δu=(u-q-\fracκ{2π}\ln\frac{1}{\ep})_+^p-(q-\fracκ{2π}\ln\frac{1}{\ep}-u)_+^p, \quad & x\inΩ, u=0, \quad & x\in\partialΩ, {cases} \] where $p>1$, $Ω\subset\mathbb{R}^2$ is a bounded domain, $q$ is a harmonic function. We showed that if $Ω$ is a simply-connected smooth domain, then for any given non-degenerate critical point of Kirchhoff-Routh function $\mathcal{W}(x_1^+,...,x_m^+,x_1^-,...,x_n^-)$ with $κ^+_i=κ>0\,(i=1,...,m)$ and $κ^-_j=-κ\,(j=1,...,n)$, there is a stationary classical solution approximating stationary $m+n$ points vortex solution of incompressible Euler equations with total vorticity $(m-n)κ$.

math.AP↗
arXiv2012

Regularization of point vortices for the Euler equation in dimension two

In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem [ -\ep^2 Δu=(u-q-\fracκ{2π}\ln\frac{1}{\ep})_+^p, \quad & x\inΩ, u=0, \quad & x\in\partialΩ, ] where $p>1$, $Ω\subset\mathbb{R}^2$ is a bounded domain, $q$ is a harmonic function. We showed that if $Ω$ is simply-connected smooth domain, then for any given non-degenerate critical point of Kirchhoff-Routh function $\mathcal{W}(x_1,...,x_m)$ with the same strength $κ>0$, there is a stationary classical solution approximating stationary $m$ points vortex solution of incompressible Euler equations with vorticity $mκ$. Existence and asymptotic behavior of single point non-vanishing vortex solutions were studied by D. Smets and J. Van Schaftingen (2010).

math.AP↗
arXiv2012

An optimal bound on the number of interior spike solutions for Lin-Ni-Takagi problem

We consider the following singularly perturbed Neumann problem {eqnarray*} \ve^2 Δu -u +u^p = 0 \quad {in} \quad Ω, \quad u>0 \quad {in} \quad Ω, \quad {\partial u \over \partial ν}=0 \quad {on} \quad \partial Ω, {eqnarray*} where $p$ is subcritical and $Ω$ is a smooth and bounded domain in $\R^n$ with its unit outward normal $ν$. Lin-Ni-Wei \cite{LNW} proved that there exists $\ve_0$ such that for $0<\ve<\ve_0$ and for each integer $k$ bounded by {equation} 1\leq k\leq \frac{δ(Ω,n,p)}{(\ve |\log \ve |)^n} {equation} where $δ(Ω,n,p)$ is a constant depending only on $Ω$, $p$ and $n$, there exists a solution with $k$ interior spikes. We show that the bound on $k$ can be improved to {equation} 1\leq k\leq \frac{δ(Ω,n,p)}{\ve^n}, {equation} which is optimal.

math.AP↗
arXiv2012

On Entire Solutions of an Elliptic System Modeling Phase Separations

We study the qualitative properties of a limiting elliptic system arising in phase separation for Bose-Einstein condensates with multiple states: Δu=u v^2 in R^n, Δv= v u^2 in R^n, u, v>0\quad in R^n. When n=1, we prove uniqueness of the one-dimensional profile. In dimension 2, we prove that stable solutions with linear growth must be one-dimensional. Then we construct entire solutions in $\R^2$ with polynomial growth $|x|^d$ for any positive integer $d \geq 1$. For $d\geq 2$, these solutions are not one-dimensional. The construction is also extended to multi-component elliptic systems.

math.AP↗
arXiv2012

The Stability of Steady-State Hot-Spot Patterns for a Reaction-Diffusion Model of Urban Crime

The existence and stability of localized patterns of criminal activity are studied for the reaction-diffusion model of urban crime that was introduced by Short et. al. [Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp. 1249--1267]. Such patterns, characterized by the concentration of criminal activity in localized spatial regions, are referred to as hot-spot patterns and they occur in a parameter regime far from the Turing point associated with the bifurcation of spatially uniform solutions. Singular perturbation techniques are used to construct steady-state hot-spot patterns in one and two-dimensional spatial domains, and new types of nonlocal eigenvalue problems are derived that determine the stability of these hot-spot patterns to ${\mathcal O}(1)$ time-scale instabilities. From an analysis of these nonlocal eigenvalue problems, a critical threshold $K_c$ is determined such that a pattern consisting of $K$ hot-spots is unstable to a competition instability if $K>K_c$. This instability, due to a positive real eigenvalue, triggers the collapse of some of the hot-spots in the pattern. Furthermore, in contrast to the well-known stability results for spike patterns of the Gierer-Meinhardt reaction-diffusion model, it is shown for the crime model that there is only a relatively narrow parameter range where oscillatory instabilities in the hot-spot amplitudes occur. Such an instability, due to a Hopf bifurcation, is studied explicitly for a single hot-spot in the shadow system limit, for which the diffusivity of criminals is asymptotically large. Finally, the parameter regime where localized hot-spots occur is compared with the parameter regime, studied in previous works, where Turing instabilities from a spatially uniform steady-state occur.

nlin.PS↗
arXiv2011

Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources

We consider the following Toda system Δu_i + \D \sum_{j = 1}^n a_{ij}e^{u_j} = 4πγ_{i}δ_{0} \text{in}\mathbb R^2, \int_{\mathbb R^2}e^{u_i} dx < \infty, \forall 1\leq i \leq n, where $γ_{i} > -1$, $δ_0$ is Dirac measure at 0, and the coefficients $a_{ij}$ form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result: $$\sum_{j=1}^n a_{ij}\int_{\R^2}e^{u_j} dx = 4π(2+γ_i+γ_{n+1-i}), \;\;\forall\; 1\leq i \leq n.$$ This generalizes the classification result by Jost and Wang for $γ_i=0$, $\forall \;1\leq i\leq n$. (ii) We prove that if $γ_i+γ_{i+1}+...+γ_j \notin \mathbb Z$ for all $1\leq i\leq j\leq n$, then any solution $u_i$ is \textit{radially symmetric} w.r.t. 0. (iii) We prove that the linearized equation at any solution is \textit{non-degenerate}. These are fundamental results in order to understand the bubbling behavior of the Toda system.

math.AP↗
arXiv2011

Non-compactness of the Prescribed Q-curvature Problem in Large Dimensions

Let $(M, g)$ be a compact Riemannian manifold of dimension $N \geq 5$ and $Q_g$ be its $Q$ curvature. The prescribed $Q$ curvature problem is concerned with finding metric of constant $Q$ curvature in the conformal class of $g$. This amounts to finding a positive solution to \[ P_g (u)= c u^{\frac{N+4}{N-4}}, u>0 {on} M\] where $P_g$ is the Paneitz operator. We show that for dimensions $N \geq 25$, the set of all positive solutions to the prescribed $Q$ curvature problem is {\em non-compact}.

math.DG↗
arXiv2011

New entire positive solution for the nonlinear Schrodinger equation: Coexistence of fronts and bumps

In this paper we construct a new kind of positive solutions of $$\De u-u+u^{p}=0 \text{on} \R^2$$ when $p> 2.$ These solutions $\displaystyle{u(x,z)\sim \om(x-f(z))+ \sum_{i=1}^{\infty}\om_{0}((x, z)-ξ_i\vec{e}_{1})}$ as $L\rightarrow +\infty$ where $\om$ is a unique positive homoclinic solution of $\om"-\om+\om^{p}=0$ in $\R$ ; $\om_{0}$ is the two dimensional positive solution and $\vec{e}_{1}= (1, 0)$ and $ξ_{j}$ are points such that $ξ_{j}= jL+ \mathcal{O}(1)$ for all $j\geq 1.$ This represents a first result on the {\em coexistence} of fronts and bumps. Geometrically, our new solutions correspond to {\em triunduloid} in the theory of CMC surface.

math.AP↗
arXiv2011

The Lin-Ni's problem for mean convex domains

We prove some refined asymptotic estimates for postive blowing up solutions to $Δu+εu=n(n-2)u^{\frac{n+2}{n-2}}$ on $Ω$, $\partial_νu=0$ on $\partialΩ$; $Ω$ being a smooth bounded domain of $\rn$, $n\geq 3$. In particular, we show that concentration can occur only on boundary points with nonpositive mean curvature when $n=3$ or $n\geq 7$. As a direct consequence, we prove the validity of the Lin-Ni's conjecture in dimension $n=3$ and $n\geq 7$ for mean convex domains and with bounded energy. Recent examples by Wang-Wei-Yan show that the bound on the energy is a necessary condition.

math.AP↗