arXiv subjects
Publications and source records attributed to Juncheng Wei.
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 $\R^2$.
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.
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.
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 $|x|^d$.
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.
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 $(m-n)κ$.
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).
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.
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.
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.
We give a new bound on the exponent for the nonexistence of stable solutions to the biharmonic problem $$Δ^2u=u^p,\quad u>0 in \mathbb{R}^n $$ where $p>1, n \geq 20$.
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.
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}.
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.
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.
In this paper, we are interested in bounded, entire, solutions of the Allen-Cahn equation which are defined in Euclidean space and whose zero set is asymptotic to a given minimal cone. In particular, in dimension larger than or equal to 8, we prove the existence of stable solutions of the Allen-Cahn equation whose zero sets are not hyperplanes.
We consider the following semilinear elliptic equation on a strip: \[ \left\{{array}{l} Δu-u + u^p=0 \ {in} \ \R^{N-1} \times (0, L), u>0, \frac{\partial u}{\partial ν}=0 \ {on} \ \partial (\R^{N-1} \times (0, L)) {array} \right.\] where $ 1< p\leq \frac{N+2}{N-2}$. When $ 1 0$ such that for $L \leq L_{*}$, the least energy solution is trivial, i.e., doesn't depend on $x_N$, and for $L >L_{*}$, the least energy solution is nontrivial. When $N \geq 4, p=\frac{N+2}{N-2}$, it is shown that there are two numbers $L_{*} L_{**}$. A connection with Delaunay surfaces in CMC theory is also made.
We study the orbital stability and instability of single-spike bound states of semiclassical nonlinear Schrodinger (NLS) equations with critical exponent, linear and nonlinear optical lattices (OLs). These equations may model two-dimensional Bose-Einstein condensates in linear and nonlinear OLs. When linear OLs are switched off, we derive the asymptotic expansion formulas and obtain necessary conditions for the orbital stability and instability of single-spike bound states, respectively. When linear OLs are turned on, we consider three different conditions of linear and nonlinear OLs to develop mathematical theorems which are most general on the orbital stability problem.