arXiv subjects

Juncheng Wei

Publications and source records attributed to Juncheng Wei.

At least 19 recordsLinked to original sources

arXiv2026

Planar Gross--Pitaevskii traveling waves at every subsonic speed

For every subsonic speed $c\in(0,\sqrt2)$, we prove the existence of a finite-energy traveling wave for the planar Gross--Pitaevskii equation. This resolves the longstanding problem of the existence of prescribed-speed traveling-wave solutions in two dimensions, explicitly stated as open by Mari\c{s} (Ann. of Math., 2013) and Bellazzini and Ruiz (Amer. J. Math., 2023). The proof relies essentially on the energy estimate \begin{equation*} E(\psi)\le C_J\bigl(I_c(\psi)+\ind(\psi)\bigr), \qquad c\in J, \end{equation*} where $E$ is the energy, $I_c$ the action at speed $c$, $\ind$ the real Morse index, $J$ is any compact interval contained in $(0,\sqrt2)$, and $C_J$ is a positive constant depending only on $J$. We also prove finite-bubble compactness, including splitting of the energy, action, potential energy, and momentum, and attainment of the action among nonconstant waves of Morse index at most one.

math.AP↗

arXiv2026

A Bernstein problem for translating solutions to the mean curvature flow

We study entire graphical translating solutions of the mean curvature flow, \[ \operatorname{div}\!\left(\frac{\nabla G}{\sqrt{1+|\nabla G|^2}}\right) =\frac{1}{\sqrt{1+|\nabla G|^2}} \qquad\text{in }\mathbb R^N. \] Every such graph is mean-convex, since its mean curvature is the vertical component of its unit normal. In dimension two, mean-convex translating solitons are convex, and an entire graphical translator is therefore the rotationally symmetric bowl soliton. In higher dimensions Wang constructed non-rotational entire convex translating graphs. We prove that a further loss of rigidity occurs at the Bernstein dimension: for every $N\ge 8$ there exists a one-parameter family of entire graphical translators that are mean-convex but not convex. The construction starts from the Bombieri--De Giorgi--Giusti (BDG) entire minimal graph in $\mathbb R^8$ and develops a singular perturbation theory for the translator equation around it. The main new feature is a transition layer near Simons' cone: the translating term breaks the odd symmetry of the minimal graph, and after a suitable recentering the matching problem is governed by a parabolic inner equation. A detailed analysis of this layer, together with weighted Jacobi theory on the BDG graph and global barriers, yields the desired entire solutions.

math.DG↗

arXiv2026

Entire monotone solutions of the anisotropic Allen-Cahn equation in dimension 5

In this paper, we consider the anisotropic Allen-Cahn equation $-\operatorname{div} a(Du)+W'(u)=0$ in $\mathbb{R}^N$, where $a(p):=DH(p)$ with $H(p)=\frac{1}{2}F(p)^2$ and $F$ a uniformly elliptic integrand, and $W(u)=\frac{1}{4}(1-u^2)^2$. Based on the Mooney-Yang anisotropic minimal graph, we prove that the anisotropic Allen-Cahn equation admits a stable solution for $N\geq4$ in the weak sense, whose level sets are not hyperplanes. As a byproduct, we also construct a smooth solution of the above anisotropic Allen-Cahn equation for $N\geq5$ that is monotone in one direction but is not one-dimensional.

math.AP↗

arXiv2026

A counterexample to a strong maximum principle for the sixth-order GJMS operator

We exhibit an explicit closed seven-dimensional Riemannian manifold \[ (M,g)=\mathbb S^2(1)\times \mathbb S^5\left(\frac1{100}\right), \] where the displayed parameters denote sectional curvatures, for which $\Ric_g>0$, and hence $Q_g^{(2)}>0$. Moreover, \[ Q^{(4)}_g>0,\qquad Q^{(6)}_g>0, \] and the sixth-order GJMS operator $P_{6,g}$ is strictly positive as a self-adjoint operator, but nevertheless $P_{6,g}$ fails the strong maximum principle. The failure is caused by a nonconstant positive eigenvalue of $P_{6,g}$ lying strictly below the eigenvalue of the constant mode. The example also has $Y_2(M,[g])>0$ and $Y_4(M,[g])>0$, while $P_{6,g}$ does not have a positive Green function. It disproves Conjecture~1 of Andrade, Piccione, and Wei and its general-order formulation by Case and Gover.

math.DG↗

arXiv2026

On Brezis' open problem 2.2

We prove that the global minimizer of the Ginzburg-Landau energy in the disk of radius $R$ with boundary value $ u(x)=\frac{x}{|x|}$ is the degree-one radial solution of the planar Ginzburg--Landau equation. This gives an affirmative answer to Open Problem~2.2 in Brezis' open-problem list. This is achieved by comparing the radial solution $f$ in the disk with the degree-one radial solution $F$ in the whole plane. Multiplying a disk competitor by $F/f$ enables us to use the known minimality of the whole-plane vortex without changing the boundary trace. The difference of the two energies can be decomposed into Fourier modes. Every nonzero mode is nonnegative, and the zero mode is then handled by a Picone type identity.

math.AP↗

arXiv2026

A positive answer to the generalized Chang-Yang conjecture on $\mathbb{S}^N$

We prove that for every integer $N\geq 3$ and $\alpha\geq \frac{1}{2}$, Beckner's inequality \[ \frac{\alpha}{2}\int_{\mathbb{S}^N}u(P_{N}u) dw+(N-1)!\int_{\mathbb{S}^N}u dw-\frac{(N-1)!}{N}\log\int_{\mathbb{S}^N}e^{Nu} dw\geq 0 \] holds for every $u\in H^{\frac{N}{2}}(\mathbb{S}^N)$ whose center of mass is at the origin. The proof is mainly based on an integral representation formula and a rigidity theorem for stable critical points. Hence, we answer the generalized Chang-Yang conjecture positively for every integer $N\geq 3$.

math.AP↗

arXiv2026

Nonradial stable solutions near the Joseph--Lundgren threshold

We study positive stable solutions of the supercritical Lane--Emden equation in the first Joseph--Lundgren interval. For a family of dimensions, we construct nonradial stable entire solutions with exponent close to the upper endpoint of this interval. This disproves a radiality conjecture of Chan and Wei. The construction begins with a smooth positive nonconstant solution on the sphere, which is obtained by matching a polar cap to an inner neck. A sharp expansion of the lowest shifted eigenvalue proves that the resulting singular cone is strictly stable. Finally, a minimal-solution and rescaling argument replaces the cone by a smooth stable entire solution while preserving its sphere variation. To the best of our knowledge, this is the first nontrivial example of nonradial stable solutions for the Emden-Fowler equation.

math.AP↗

arXiv2026

Sharp Beckner's Inequalities for Axially Symmetric Functions on $\mathbb{S}^N$

We prove that for every integer $N\geq 3$ and $\alpha\geq \frac{1}{2}$, Beckner's inequality \begin{equation*} \frac{\alpha}{2}\int_{\mathbb{S}^N}u(P_{N}u) dw+(N-1)!\int_{\mathbb{S}^N}u dw-\frac{(N-1)!}{N}\ln\int_{\mathbb{S}^N}e^{Nu} dw\geq 0 \end{equation*} holds for any axially symmetric $u\in H^{\frac{N}{2}}(\mathbb{S}^N)$ whose center of mass is at the origin. The proof is mainly based on a weighted $\ell ^2$ estimate on Gegenbauer coefficients and a rigidity theorem for stable critical points. Hence, we answer the generalized Chang-Yang conjecture positively in the axially symmetric case for every integer $N\geq 3$.

math.AP↗

arXiv2026

Existence of Yamabe stability optimizers

We prove the existence of stability optimizers for the Yamabe inequality on closed Riemannian manifolds of dimension at least three with positive Yamabe invariant that satisfy two threshold conditions. Remarkably, the compactness threshold we uncover is different from the special case of the round sphere treated previously by the second author. More precisely, it is given by sequences blowing up in one instead of two bubbles, reflecting the compactness of Yamabe minimizers in the non-spherical case. Using the classical asymptotic analysis of Aubin--Schoen test functions, we prove that the stability constant is strictly below the one-bubble threshold in dimension at least six and when the manifold is not locally conformally flat. In the complementary case, namely in dimensions three through five or when the manifold is locally conformally flat, we find a new positive-mass-type condition which is sufficient for the strict inequality.

math.DG↗

arXiv2026

Nondegeneracy and Morse Index of Ginzburg--Landau Vortices

We prove that the standard degree-two and degree-three vortex solutions of the Ginzburg-Landau equation are nondegenerate. Their Morse indices are also computed. The proof relies on new explicit upper and lower bounds of the modulus of these solutions and a comparison argument. It is expected that our method can be generalized to study higher degree solutions.

math.AP↗

arXiv2026

On Sirakov's equal-frequency uniqueness conjecture

Let $N\in\{2,3\}$, $0<\mu _1\leq\mu _2$, and $0<\beta<\mu _1$. We prove that the equal-frequency two-component cubic Schr\"odinger system \[ -\Delta u+u=\mu _1u^3+\beta uv^2, \qquad -\Delta v+v=\mu _2v^3+\beta u^2v \quad\text{in }\mathbb{R}^N \] has exactly one positive solution in $H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)$ modulo simultaneous translations. More precisely, every positive solution is a simultaneous translate of the synchronized state constructed from the unique positive radial solution of $-\Delta w+w=w^3$ in $\mathbb{R}^N$. This settles Sirakov's equal-frequency uniqueness conjecture throughout the weak-coupling range. The main difficulty in the proof is to exclude radial solutions for which the ratio of the normalized components is nonconstant. After normalization, the two components satisfy scalar equations with a common potential. We construct a weighted Pohozaev functional for the system together with a correction term and prove that both the corrected functional and the associated weighted functional are strictly positive. Combining these sign properties with a radial flux identity and an auxiliary quotient associated with the component ratio forces synchronization.

math.AP↗

arXiv2026

Qualitative bifurcation diagram for Grad-Shafranov type equations

We study the qualitative behavior of solutions of Grad-Shafranov type equations arising in plasma physics with general differential operators and general nonlinearities. In particular, we extend recent estimates about threshold values for uniqueness, monotonicity and non-existence of the free boundary. The argument is based on a refined spectral analysis for weighted non-local problems together with comparison techniques and level set analysis.

math.AP↗

arXiv2026

Finite Potential Energy for Entire Solutions of the Planar Ginzburg--Landau Equation

We prove that every smooth entire solution $ u\colon\mathbb{R}^2\to\mathbb{R}^2 $ of the Ginzburg--Landau equation $ -\Delta u=u(1-|u|^2) $ with $ |u(x)|\to1 $ as $ |x|\to\infty $ has finite potential energy, i.e., \begin{equation*} \int_{\mathbb{R}^2}(1-|u|^2)^2 \mathrm{d}x<+\infty, \end{equation*} thereby resolving Brezis' Open Problem 2.5 in [4]. The main difficulty stems from the possible presence of a curl-free mode that carries nonzero circulation and decays only like $ |x|^{-1} $; such a mode lies outside $ L^2 $ and does not admit a single-valued potential. By minimizing over $ L^2 $ gradient corrections, we construct a comparison field that solves the homogeneous equation and inherits the same circulation. The Kelvin inversion, combined with the De Giorgi--Nash--Moser theory for quasilinear elliptic equations, then produces the optimal decay $ O(|x|^{-1}) $. For a Ginzburg--Landau solution, the Bernstein estimate and the coercivity of the Jacobi form produce an $ L^2 $ forcing term in the exterior phase equation. The resulting $ L^4 $ bound on the phase field implies $1-|u|^2\in L^2(\mathbb{R}^2)$, and therefore the potential energy is finite.

math.AP↗

arXiv2026

Collapsing-tube type II blow-up for the energy-supercritical heat equation

We construct a new type II finite-time blow-up mechanism for the energy-supercritical heat equation \[ u_t=\Delta u+u^3, \qquad n\geq 5. \] The solution is positive and blows up only at the origin, but in a highly anisotropic fashion. As $t\nearrow T$, the solution concentrates in a thin tubular region around an $(n-4)$-dimensional sphere whose radius shrinks to zero at the self-similar scale \[ \xi_r(t)\sim \sqrt{2(n-4)(T-t)}. \] At the same time, concentration takes place transversely to the sphere at the much smaller scale \[ \lambda(t)\sim \kappa_* \frac{T-t}{|\log(T-t)|^{\frac n{n-2}}}, \] for some $\kappa_*>0$. More precisely, in cylindrical coordinates $r=|x'|$, $z\in\mathbb R^3$, the leading profile is \[ u(x,t) \sim \frac{1}{\lambda(t)} U\left( \frac{r-\xi_r(t)}{\lambda(t)}, \frac{z}{\lambda(t)} \right), \] where $U$ is the Aubin--Talenti bubble in $\mathbb R^4$. The construction reveals a two-scale singularity mechanism in which a critical transverse bubble concentrates around a geometric set that itself collapses. The concentration tube evolves at the parabolic scale $\sqrt{T-t}$, whereas its transverse thickness is governed by the much smaller type II scale $\lambda(t)$. The logarithmic blow-up law is determined by a nonlocal modulation equation arising from the interaction between the four-dimensional critical bubble and the axisymmetric heat kernel. To our knowledge, this seems to be the first Type II blowup that quantifies the effect of a self-similar collapsing tube. The exponent $p=3$ is energy-supercritical in dimensions $n\geq5$, but lies below the Joseph--Lundgren exponent for $5\leq n\leq 12$, in a regime where positive radial type II blow-up is ruled out. The present result provides the first example of a positive type II, single-point blow-up through a collapsing thin-tube geometry.

math.AP↗

arXiv2026

Giga-Kohn-type results for the fully fractional heat equation

We consider the semilinear fully fractional heat equation \[ (\partial_t-\Delta)^\sigma u = |u|^{p-1}u \quad \text{in } \mathbb{R}^n \times \mathbb{R}_{-}, \qquad 0 < \sigma < 1. \] For $n\leq 2\sigma$ or $1<p\leq \frac{n+2\sigma}{n-2\sigma}$, we generalize the monotonicity formula and Liouville-type theorem when $\sigma=1$ proved by Giga and Kohn. In order to overcome the difficulty that this equation is nonlocal, we give a new interpretation of the classical Giga-Kohn's Pohozaev identity in terms of Hermite expansion. This insight is new and interesting even for $\sigma=1$. We further establish a space-time nonlocal monotonicity formula for the self-similar equation. As far as we are concerned, this is the first monotonicity formula for space-time nonlocal equations without using an extension by Stinga and Torrea.

math.AP↗

arXiv2026

A refined blow-up analysis of the Brezis-Nirenberg equation and its application: The one-bubble case for $N\geq4$

In this paper, we consider the famous Brezis-Nirenberg equation \begin{eqnarray*} \left\{ \aligned &-\Delta u=\lambda u+|u|^{\frac{4}{N-2}}u,\quad&\mbox{in}\,\, \Omega,\\ &u=0,\quad&\mbox{on}\,\, \partial\Omega, \endaligned \right. \end{eqnarray*} where $N\geq3$ is the dimension, $\Omega\subset\mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$ and $\lambda>0$ is a parameter. By developing a refined blow-up analysis based on the inverse reduction argument developed in \cite{WW2019,WW2019-2}, we classify, for the fist time, the Struwe decomposition of the Brezis-Nirenberg equation in the one-bubble case as the parameter $\lambda$ varies for $N\geq4$. As applications, we prove that the $4d$ Brezis-Nirenberg equation has a nontrivial solution (least energy solution) for $\lambda\in\sigma(-\Delta)$ in general bounded domains, where $\sigma(-\Delta)$ is the spectrum of $-\Delta$ in $H^1_0(\Omega)$. Our result completes the existence theory of the Brezis-Nirenberg equation for $N\geq4$ in \cite{AP2025,CFP1985,CFS,CSS1986,CW2005,CSZ2012,SWW2009,TYZ2022} since 1984.

math.AP↗

arXiv2026

Complete Classification and Nondegeneracy of $N$-Component Cubic Nonlinear Schr\"{o}dinger System in ${\mathbb R}$

We study the one-dimensional cubic nonlinear Schr\"{o}dinger system \[ u_i''+2\left(\sum_{k=1}^N u_k^2\right)u_i=-\mu_i u_i \quad \mbox{in } \ \mathbb R,\ \ i=1,2,\cdots,N, \] where $u=(u_1,\cdots,u_N)\in \big(H^1(\mathbb{R})\big)^N$, $\mu_1\leq\mu_2\leq\cdots\leq\mu_N<0$, and $N\geq 2$ is arbitrary. In this paper, we prove the following results for any $N\ge 2$: (i). All nontrivial solutions of the system can be completely classified; (ii). The linearized operator at any nontrivial solution of the system is non-degenerate; (iii). For all $i=1, 2,\cdots, N$, the exact $L^2$-mass identity of $u_i$ is derived in terms of $2\sqrt {|\mu_i|}$, which yields a complete characterization of normalized solutions satisfying $\int_{\mathbb{R}}u_i^2dx=1$. These settle some conjectures of [R. Frank, D. Gontier and M. Lewin, CMP, 2021] and [Y. Guo, Y. Luo and J. Wei, APDE, 2026], where the system was addressed specially for $N=2$ and $N=3$, respectively.

math.AP↗

arXiv2026

On Brezis Open Problem 3.1

Let $B_1$ be the unit disk in ${\mathbb R}^2$. We consider the harmonic map equation $$ -\Delta u=|\nabla u|^2u,$$ subject to the Dirichlet boundary condition $ u(e^{i\theta})=(R\cos\theta,R\sin\theta,\sqrt{1-R^2}):=g_R$, where $0<R<1$ and $u: B_1\to {\mathbb S}^2$ is understood in the weak harmonic-map sense. In 1983, Brezis and Coron proved the existence of two explicit solutions of this nonlinear Dirichlet problem and showed that they are the unique minimizers in their respective relative homotopy classes. In this paper, we resolve a long-standing open question originally posed in their work, later posed as Open Problem 3.1 in Brezis Favorite Open Problems List. Specifically, we prove that these two explicit maps are the only weak harmonic maps with boundary trace $g_{R}$, thereby providing a definitive affirmative answer to Brezis open problem. The proof is based on a boundary rigidity argument. An auxiliary potential $X$ associated with $u$, the Pohozaev identity for the Hopf differential, and the planar isoperimetric inequality imply $$|u_r|\equiv R, \qquad u_r\cdot u_\theta\equiv0 \qquad\text{on }\partial B_1. $$ Thus the Hopf differential vanishes on the boundary and hence, by holomorphicity, on the whole disk. The problem is then reduced to the conformal case, where a stereographic-coordinate classification gives exactly the two Brezis--Coron maps.

math.AP↗