arXiv subjects
Publications and source records attributed to Juncheng Wei.
For every integer $d\ge2$ and every subsonic speed $c\in(0,1)$, we construct a nonconstant finite-energy travelling wave for the easy-plane Landau-Lifshitz equation on $\mathbb{R}^d$. This addresses the existence problem raised in the numerical work of Papanicolaou and Spathis. The profiles satisfy the travelling-wave equation in the sense of distributions and are smooth in dimension two.
We study the dependence on the dimension of stability constants for the Caffarelli--Kohn--Nirenberg inequalities, using the weighted energy distance to the manifold of optimizers. In the subcritical non-degenerate case, i.e., $b_{FS}(a) 0$ exhibits a different dimensional scale, \[ \frac{\min\{β_{\mathrm{Sob}},4a\}}{25d^2} \le c_{d,a,a}\le\frac{16a}{d^2}+O_a(d^{-3}), \] where $β_{\mathrm{Sob}}>0$ is a universal Sobolev stability constant. In the degenerate case on the Felli--Schneider curve, for fixed $a<0$, we prove \[ \frac{β_a^{FS}}{d^3} \le\widetilde c_{d,a,b_{FS}(a)} \le\frac{32|a|}{d^3}+O_a(d^{-4}), \] where $\widetilde c$ is the quartic stability constant given in [Frank-Peteranderl, CV\&PDE, 2024] and $β_a^{FS}>0$ is independent of $d$. The three cases thus have optimal dimensional orders $d^{-1}$, $d^{-2}$, and $d^{-3}$, respectively.
We show the existence of a ``nonradial" smooth entire solution of the Ginzburg--Landau equation with degree $q\in\{8,9\}$ and modulus tending to one at infinity. This answers Brezis' Open Problem 2.4 negatively. To construct the counterexample, we vary a cone parameter $q \in [8, 9]$ and keeping the degree on the cone equal to one. Several key ingredients are needed. First, we need to compute the Morse index of radial vortex solutions at $q\in \{8, 9\}$. Second, we need to show compactness. Finally, Leray-Schauder degree theory and a homotopy argument yield the desired existence result.
We determine the global minimizers of the three-dimensional Yukawa lattice energy at every fixed density. At unit covolume, BCC is the unique minimizer below a transition parameter $κ_*\approx1.7179218684699$, FCC is the unique minimizer above it, and exactly these two lattice classes minimize at the transition. Mathematically, denote the Yukawa potential and the energy per particle under this potential by \[ v_κ(r)=\frac{e^{-κr}}r,\qquad E_κ(L)=\frac12\sum_{v\in L\setminus\{0\}} \frac{e^{-κ|v|}}{|v|}. \] Then, \[ \arg\min_{|L|=1}E_κ(L)= \begin{cases} \hbox{BCC lattice},&0<κ<κ_*, \hbox{BCC or FCC lattice},&κ=κ_*, \hbox{FCC lattice},&κ>κ_*. \end{cases} \] As a consequence, the zero-screening comparison recovers the unique BCC minimum for the renormalized Coulomb energy.
We establish a sharp pointwise gradient estimate for positive entire classical solutions of the Lane--Emden system when one of the exponents lies below the Serrin exponent $n/(n-2)$. The estimate generalizes the positivity of the scalar curvature of the associated conformal metric in the biharmonic case $q=1$. The coefficient in the estimate is optimal on the critical hyperbola.
Let $B(δ)\subset\mathbb S^n$ be a geodesic ball with the round metric $\bar{g}$ of sectional curvature one, where $n\geq3$ and $0<δ<π/2$. We prove local scalar-curvature rigidity under preservation of the induced boundary metric and a one-sided lower bound for the boundary mean curvature. More precisely, every sufficiently small $C^{2,α}$ perturbation $g$ of $\bar{g}$ satisfying $R(g)\geq n(n-1)$ and $H(g)\geq(n-1)\cotδ$ is round up to a diffeomorphism fixing the boundary pointwise. Inspired by the scalar--vector--tensor decomposition used in the study of Yamabe compactness, we establish positivity of the Brendle--Marques quadratic form in the divergence-free gauge for variations preserving the induced boundary metric and satisfying the joint linearized constraints $D R_{\bar{g}}(h)=D H_{\bar{g}}(h)=0$. The hemisphere counterexamples of Brendle--Marques--Neves show that the radius range is sharp.
We prove sharp improved Sobolev inequalities on $\mathbb{S}^n$, $n\ge 4$, for functions whose critical density $|u|^{2n/(n-2)}$ has zero barycenter, without any symmetry assumption. The Dirichlet coefficient improves by the optimal factor $2^{-2/n}$, while the coefficient of the $L^2$ term remains one. Equality holds precisely for constant functions. This establishes the sharp form of Aubin's improved Sobolev inequality in every dimension $n\ge4$.
We study the one-dimensional nonlinear Schrödinger system \[ -u_i''-2\Big(\sum_{k=1}^N u_k^2\Big)^{p-1}u_i=μ_i u_i \quad \mbox{in } \ \mathbb R,\ \ i=1,2,\cdots,N, \tag{S} \] where $u=(u_1,\cdots,u_N)\in H^1(\mathbb{R})^N$, $μ_1\leqμ_2\leq\cdots\leqμ_N<0$, $1 0$ such that $J_N(p)$ is attained for any $p\in(2-δ_N,2)$, whereas $J_N(p)$ is not attained for any $p\in[2,2+δ_N)$. These settle some conjectures of [Gontier--Lewin--Nazar, ARMA, 2021] and [Frank--Gontier--Lewin, CMP, 2021].
We prove sharp residual stability estimates for the critical Poincaré-Sobolev equation near sums of one positive hyperbolic solution and one projected Euclidean bubble in dimensions at least four. The estimates are uniform in the relative positions of the centers. The sharp rate is linear except in dimension six, where it contains a square root of a logarithm. For nonnegative functions, the same estimates hold under an energy constraint near the sum of the two profile energies for \(λ\) outside a finite set, and an additional mass constraint removes this restriction. For arbitrary finite numbers of profiles, we identify and realize all nonnegative Palais-Smale levels. We characterize the parameters at which different pairs of profile numbers have the same energy and show that the limiting energy and mass determine both profile numbers.
We prove sharp improved Sobolev inequalities on $\mathbb S^n$, $n\ge3$, for antipodally even functions and for axially symmetric functions whose critical density $|u|^{2n/(n-2)}$ has zero barycenter. In both cases, the Dirichlet coefficient improves by the optimal factor $2^{-2/n}$, while the coefficient of the $L^2$ term remains one. Equality holds precisely for constant functions. The even case resolves a conjecture of Dolbeault (2021).
We determine the sharp compactness ranges for the scalar-flat and minimal-boundary Yamabe problems on smooth compact manifolds of positive conformal type, excluding the conformal round hemisphere. For zero scalar curvature and positive constant boundary mean curvature, compactness holds through dimension $14$ for general boundary and dimension $21$ for umbilic boundary. For positive scalar curvature and zero boundary mean curvature, the corresponding upper dimensions are $14$ and $20$. Together with the noncompactness examples in Part I, these results identify the transition dimensions in both boundary classes. For positive scalar curvature, we also prove compactness through dimension eight for every fixed real boundary mean curvature, and obtain higher-dimensional ranges when this curvature is near zero or sufficiently large and positive. The proof combines scalar-correction estimates for the full conformal Fermi metric expansion with a geometric formula expressing the logarithmic coefficient of the corrected energy as a negative sum of squares.
For $n\ge3$ and $γ>-1$, we study normal solutions of \begin{equation*} (-Δ)^{n/2}u=(n-1)!\,|x|^{nγ}e^{nu} \quad\hbox{in }\R^n. \end{equation*} We reformulate this problem on the cylinder via the Emden--Fowler transform. We first establish Kelvin symmetry for normal solutions. We then prove existence and uniqueness, up to dilation, of the radial normal solution and establish its radial nondegeneracy. Next, we carry out a detailed spectral analysis of the linearized operator and construct branches of nonradial normal solutions by the Crandall--Rabinowitz theorem. Finally, we prove that every normal solution is radial when $γ>0$ is sufficiently small.
We prove that for every $N\geq 3$, every solution to the $Q$-curvature-type problem $$ αP_N u + (N-1)!\left(1-\frac{e^{Nu}}{\int_{\mathbb{S}^N} e^{Nu}dw}\right)=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^N $$ is constant, provided that $ α\ge\frac{1}{2}$ and $α\not =1$. The proof consists of a spectral gap estimate and pointwise estimates of the normalized density $\left(\int_{\mathbb{S}^N} e^{Nu}dw\right)^{-1}e^{Nu}$.
For $s>0$, and $z\in\mathbb H=\{z=x+iy:y>0\}$, define $θ(s;z)=\sum_{m,n\in\mathbb Z}e^{-πs\big(m^2y+\frac{(mx-n)^2}{y}\big)}$ and $ J(z;a,b)=\sum_{m,n\in\mathbb Z}e^{-π\big(m^2y+\frac{(mx-n)^2}{y}\big)} \cos\bigl(2π(ma+nb)\bigr)$ for the classical and shifted theta functions (Gaussian lattice sums), respectively. We prove that, up to the modular group, there exist three thresholds $0<α_a<α_b<α_c<1$ such that \begin{equation}\aligned\nonumber \Minima_{z\in\mathbb H,\,(a,b)\in\mathbb R^2}\Big(θ(1;z)+αJ(z;a,b)\Big)= \begin{cases} \;\big(e^{iπ/{3}};1/3,1/3\big), &\hbox{if}\;\; α\in[0,α_a),\\ \;\big(e^{iπ/{3}};1/3,1/3\big)\;\hbox{or}\;\big(e^{i{φ_{α_a}}};1/2,1/2\big), &\hbox{if}\;\; α=α_a,\\ \;\big(e^{i{φ_α}};1/2,1/2\big),\;φ_α\in(φ_{α_a},\fracπ{2}),\;\; &\hbox{if}\;\; α\in(α_a,α_b),\\ \;\big(i;1/2,1/2\big),\; &\hbox{if}\;\; α\in[α_b,α_c],\\ \;\big(iy_α;1/2,1/2\big),\; y_α\in(1,\sqrt3],\;\;y_1=\sqrt3,\;\;&\hbox{if}\;\; α\in(α_c,1], \end{cases} \endaligned\end{equation} where the complex variable $z$ and vector $(a,b)$, represent the total vortex shapes and relative positions of the two-component Bose gas, respectively. Consequently, this completely proves the conjecture of Mueller and Ho~\cite{Mue2002} (2002) arising in Bose--Einstein condensates.
Let $n\ge2$, $0 0$. We prove a Liouville theorem for the fractional Lane--Emden system \[ (-Δ)^s u=v^p,\qquad (-Δ)^s v=u^q, \qquad u, v>0 \qquad\text{in }\R^n. \] More precisely, we show that the system does not have a strictly positive entire solution whenever \[ \frac1{p+1}+\frac1{q+1}>\frac{n-2s}{n}. \] No radial symmetry, global boundedness, finite-energy assumption, or prescribed decay at infinity is required. Our proof adapts the localized potential-kernel virial and source-layer interval-pressure method recently introduced for the classical Hénon--Lane--Emden system. The basic reason the method also works in the fractional setting is simple: the Riesz kernel associated with $(-Δ)^s$ has the same logarithmic derivative structure as the Newton kernel, with $n-2$ replaced by $n-2s$. However, two genuinely nonlocal points require special treatments. When $2s\le1$, the individual kernel-gradient terms are not locally absolutely integrable, so the two equations must be combined before the diagonal limit is taken. When $2s>n-1$, the kernel restricted to a line is not integrable at infinity, and the interval comparison must therefore be formulated using nonnegative extended integrals. After these issues are resolved, a localized virial identity, a one-dimensional interval-pair inequality, and an averaged source-layer estimate yield a universal local energy bound. The subcritical scaling then forces the same energy to diverge, giving the desired contradiction.
This work is devoted to the study of blow-up phenomena for solutions of \[-Δu=λV(x)|x|^{2N}e^v,\quad \mbox{in}\quad B_1,\quad v=0\quad \mbox{on}\quad \partial B_1. \] where $B_1$ denotes the unit ball in $\mathbb R^2$ centered at the origin, $V(x)$ is a positive smooth function, $N\in \mathbb N$ and $λ>0$ is a small parameter. The main goal is to determine conditions on the potential $V$ guaranteeing the existence of families of solutions that blow up at the origin as $λ\to 0^+$. Moreover, we establish higher order criteria that distinguish between simple and non-simple blow-up behavior at $x=0$. In particular, for $N-1\in 2^{\mathbb N}$, we construct a non-simple blow-up sequence $v_k$, corresponding to coefficient function $V_k$ whose derivatives vanish up to the maximal order allowed by the upper bound of Wei-Zhang, thereby proving that this bound is sharp.
For every integer $N\ge4$, we construct a fixed smooth, non-locally-conformally-flat metric on the $n$-dimensional unit sphere $\mathbb{S}^n$ for which the constant $Q_{2N}$-curvature equation admits an $L^\infty$-unbounded sequence of positive solutions. The construction applies for $n\ge2N+20$ when $N=4,5$, $n\ge2N+19$ when $6\le N\le8$, $n\ge2N+18$ when $9\le N\le17$, and $n\ge2N+17$ when $N\ge18$. For each $L \in \mathbb{N} \cup \{0\}$, the metric can be chosen arbitrarily close to the round metric in the $C^L$ norm. Together with the known cases $N=1,2,3$, this establishes noncompactness at every order, with dimension bounds that we expect to be optimal. We derive an explicit formula for the fixed-volume Hessian of total $Q_{2N}$-curvature in transverse-traceless (TT) directions at closed Einstein metrics satisfying $\operatorname{Ric}=(n-1)g$. Building on Juhl's formulas, we identify this Hessian, for every $N\in\mathbb{N}$ and $n>2N$, as a degree-$N$ polynomial in the Lichnerowicz Laplacian. For the metric perturbations generated by algebraic Weyl tensors in the construction, the quadratic reduced energy with fixed bubble center is a positive multiple of this Hessian restricted to a four-dimensional TT space. Differentiation with respect to the bubble scale then yields a finite matrix whose exact sign analysis establishes the required indefiniteness in the stated dimension ranges. After continuation to real $n$, this matrix changes from negative definite to indefinite at $n_N=2N+a_*+c_*N^{-1}+O(N^{-2})$ as $N\to\infty$, with $a_*\approx16.201871$ and $c_*\approx13.512880$, explaining the eventual bound $n\ge2N+17$.
We prove a sheeting theorem for stable solutions to the Allen--Cahn equation in ambient dimensions $2$ through $10$. We show that weak convergence of the measures $\varepsilon|Du_\varepsilon|^2\,dX$ to a hyperplane with integer multiplicity implies that the zero sets are disjoint smooth graphs. As a consequence, stable entire solutions in $\mathbb R^N$, $N\le7$, with $O(R^{N-1})$ energy growth are one-dimensional. When $W''(-1)=W''(1)$, every blowdown of an entire solution with finite Morse index and this energy growth has multiplicity one for $4\le N\le10$. For even potentials, we also obtain multiplicity one for limits with bounded energy and Morse index on closed manifolds of dimensions $3$ through $7$, under bumpiness or positive Ricci curvature.