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Zhongwei Tang

Publications and source records attributed to Zhongwei Tang.

At least 19 recordsLinked to original sources
arXiv2026

A Sharp Sobolev Inequality on Compact CR Manifolds

For any \(n\geq 1\), let \((M,θ)\) be a compact strictly pseudoconvex hypersurface type CR manifold with dimension \(2n+1\), \(Q=2n+2\) be its homogeneous dimension. We prove the sharp Folland--Stein Sobolev inequality on \(M\): if \(Q^*=2Q/(Q-2)\), then there exists a constant \(A>0\), depending only on \((M,θ)\), such that \[ \|u\|_{L^{Q^*}(M)}^2 \le K(n)^2\|\nabla_b u\|_{L^2(M)}^2 + A\|u\|_{L^2(M)}^2, \qquad u\in S^{1,2}(M). \] Here \(K(n)\) is the sharp constant in the Folland--Stein Sobolev inequality on the Heisenberg group \(\mathbb H^n\).

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arXiv2026

Blow-up analysis and compactness for critical Dirac equations on $\mathbb S^n$

We study blow-up and compactness for the critical nonlinear Dirac equation \[ D_{g_0}ψ_i = H_i|ψ_i|^{\frac{2}{n-1}}ψ_i \qquad \text{on }\mathbb S^n,\quad n\ge 3, \] where $g_0$ is the round metric and $H_i\to H>0$ in $C^0(\mathbb S^n)$. Under a first-threshold bound on the critical mass, we prove that every blowing-up sequence has, after passing to a subsequence, a unique blow-up point $p$ satisfying $H(p)=\max_{\mathbb S^n}H$. The rescaled solutions converge to a ground-state Dirac bubble that carries the entire limiting critical mass. Under local uniform $C^1$ bounds on $H_i$, we use the Kelvin transform and a second rescaling to prove that $p$ is an isolated simple blow-up point. We also derive sharp pointwise and gradient estimates in cylindrical coordinates and identify the normalized Green limit. If the coefficients converge in $C^2$ near $p$, the Pohozaev identity yields $\operatorname{Hess}_{g_0}H(p)=0$. Consequently, if a positive $H\in C^2(\mathbb S^n)$ satisfies $Δ_{g_0}H(p)<0$ at every global maximum, then solutions with critical mass bounded by a level slightly above the first threshold are uniformly bounded in $L^\infty(\mathbb S^n)$. This bound remains uniform under small $C^2$ perturbations of $H$.

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arXiv2026

Sharp Signed Mass and Rigidity for Critical Finsler--Liouville Equations

We study bounded-above entire solutions of critical Finsler--Liouville equations with a possibly sign-changing prescribed function and finite absolute mass. Under a lower cone-growth condition, we establish a sharp signed-mass bound without upper growth or monotonicity assumptions. With an additional growth condition, annular compactness for a sequence of dilations yields Pohozaev boundary limits and rigidity for radially nonincreasing prescribed functions. Explicit examples show that the rigidity threshold is optimal. In the Euclidean plane, the mass estimate also gives a sharp obstruction to completeness.

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arXiv2026

Fixed-spinor radial solutions of a perturbed Euclidean spinorial Yamabe equation

We study the perturbed Euclidean spinorial Yamabe equation \[ D_{g_{\mathbb R^n}}ψ = \frac n2\bigl(1+\varepsilon K(x)\bigr) |ψ|^{2^*-2}ψ \qquad\text{on }\mathbb R^n,\quad n\ge2, \]where $2^*:=2n/(n-1)$, $\varepsilon>0$ is a small parameter and $K$ is a prescribed radial function. We establish a finite-dimensional reduction within a radial spinor space determined by a fixed unit spinor, reducing the problem to the analysis of a function of the dilation parameter. Under suitable assumptions on $K$, we obtain explicit existence criteria for solutions with this prescribed spinorial structure, together with a multiplicity criterion yielding solutions associated with distinct scales. These criteria are expressed in terms of the behavior of $K$ at the origin and weighted integral quantities.

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arXiv2026

Topological degree for singular $N$-mean field equations

We determine the Leray--Schauder degree of the singular Dirichlet $N$-mean field equation on a smooth bounded domain in $\mathbb{R}^N$, $N\geq2$, for every positive nonresonant parameter and singular exponents $-1<α_i\leq0$. The formula depends only on the Euler characteristic of the domain and the weights $(1+α_i)^{N-1}$. We also identify the homotopy type of every sufficiently negative variational sublevel with the corresponding weighted barycenter space. We prove quantization of interior concentration masses and exclude boundary concentration and residual diffuse mass. The degree computation uses common descent fields for nearby parameters, bounds for their trajectories, and a finite-dimensional comparison with the Euler characteristic of negative sublevels. Annular averaging and dilation provide local contractions, which are extended to establish the sublevel homotopy equivalence. The arguments apply to the nonlinear, possibly degenerate $N$-Laplacian with singular weights, without requiring a global Palais--Smale condition or isolation and nondegeneracy of solutions.

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arXiv2026

The CR Nirenberg Problem, Part II: Existence and Topological Degree

This paper is a continuation of Part~I \cite{QTWPartI}, where blow-up analysis and compactness results for the CR Nirenberg problem were established. We prove some existence and degree results for the CR Nirenberg problem on the standard sphere. More precisely, we first prove existence results under antipodal and, more generally, finite-group symmetries. Without symmetry, we derive a Leray--Schauder degree formula under natural flatness conditions, yielding existence results for positive prescribed curvatures with certain homogeneous expansions at critical points. We also obtain Aubin-type inequalities under a vanishing first-moment condition, uniformly near the critical exponent.

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arXiv2026

The CR Nirenberg Problem, Part I: Blow-up Analysis and Compactness

We establish a priori estimates and compactness results for positive solutions of the CR Nirenberg problem under natural weighted flatness conditions. The proofs combine refined blow-up analysis, Pohozaev identities, and the complete classification of positive entire solutions to the critical CR Yamabe equation recently obtained by Li--Liu--Ma \cite{LLM2026} and Li \cite{L2026}. Existence and degree results are established in our subsequent paper \cite{QTWPartII}.

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arXiv2026

Compactness of Solutions to Sub-Elliptic Equations with Potential on the Heisenberg Group

In this paper, we investigate the compactness of nonnegative solutions to a critical sub-elliptic equation with a nonnegative potential on the Heisenberg group. We establish compactness in all dimensions. Moreover, we show that if a sequence of solutions blows up, both the potential and its sub-Laplacian must vanish at the blow-up point. Our analysis overcomes the inherent geometric and analytical challenges posed by the Heisenberg group, including the degeneracy of the sub-Laplacian, its non-commutative structure, and the anisotropic dilation symmetry.

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arXiv2024

New type of bubbling solutions to a critical fractional Schrödinger equation with double potentials

In this paper, we study the following critical fractional Schrödinger equation: \begin{equation} (-Δ)^s u+V(|y'|,y'')u=K(|y'|,y'')u^{\frac{n+2s}{n-2s}},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{n-3}, \qquad(0.1)\end{equation} where $n\geq 3$, $s\in(0,1)$, $V(|y'|,y'')$ and $K(|y'|,y'')$ are two bounded nonnegative potential functions. Under the conditions that $K(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$, $K(r_0,y_0'')>0$ and $V(r_0,y_0'')>0$, we prove that equation (0.1) has a new type of infinitely many solutions that concentrate at points lying on the top and the bottom of a cylinder. In particular, the bubble solutions can concentrate at a pair of symmetric points with respect to the origin. Our proofs make use of a modified finite-dimensional reduction method and local Pohozaev identities.

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arXiv2024

On the CR Nirenberg problem: density and multiplicity of solutions

We prove some results on the density and multiplicity of positive solutions to the prescribed Webster scalar curvature problem on the $(2n+1)$-dimensional standard unit CR sphere $(\mathbb{S} ^{2n+1},θ_0)$. Specifically, we construct arbitrarily many multi-bump solutions via the variational gluing method. In particular, we show the Webster scalar curvature functions of contact forms conformal to $θ_0$ are $C^{0}$-dense among bounded functions which are positive somewhere. Existence results of infinitely many positive solutions to the related equation $-Δ_{\mathbb{H}} u=R(ξ) u^{(n+2) /n}$ on the Heisenberg group $\Hn $ with $R(ξ)$ being asymptotically periodic with respect to left translation are also obtained. Our proofs make use of a refined analysis of bubbling behavior, gradient flow, Pohozaev identity, as well as blow up arguments.

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arXiv2023

On the density and multiplicity of solutions to the fractional Nirenberg problem

This paper is devoted to establishing some results on the density and multiplicity of solutions to the fractional Nirenberg problem which is equivalent to studying the conformally invariant equation $P_σ(v)=K v^{\frac{n+2σ}{n-2σ}}$ on the standard unit sphere $(\mathbb{S}^n,g_0)$ with $σ\in (0,1)$ and $n\geq 2$, where $P_σ$ is the intertwining operator of order $2σ$ and $K$ is the prescribed curvature function. More specifically, by using the variational gluing method, refined analysis of bubbling behavior, extension formula, as well as the blow up analysis arguments, we obtain the existence of infinitely many multi-bump solutions. In particular, we show the smooth curvature functions of metrics conformal to $g_0$ are dense in the $C^{0}$ topology. Moreover, the related fractional Laplacian equations $(-Δ)^σ u=K(x) u^{\frac{n+2σ}{n-2σ}}$ in $\mathbb{R}^n$, with $K(x)$ being asymptotically periodic in one of the variables, are also studied and infinitely many solutions are obtained under natural flatness assumptions.

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arXiv2023

Multiple solutions for a fractional Choquard problem with slightly subcritical exponents on bounded domains

This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of the convolution type nonlinearity tends to the fractional critical one in the sense of Hardy-Littlewood-Sobolev inequality, we obtain the existence of multiple positive solutions via Lusternik-Schnirelmann category and nonlocal global compactness. Moreover, we prove that the topology of the domain furnishes a lower bound for the number of positive solutions.

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arXiv2023

Interior estimates of derivatives and a Liouville type theorem for Parabolic $k$-Hessian equations

In this paper, we establish the gradient and Pogorelov estimates for $k$-convex-monotone solutions to parabolic $k$-Hessian equations of the form $-u_tσ_k(λ(D^2u))=ψ(x,t,u)$. We also apply such estimates to obtain a Liouville type result, which states that any $k$-convex-monotone and $C^{4,2}$ solution $u$ to $-u_tσ_k(λ(D^2u))=1$ in $\mathbb{R}^n\times(-\infty,0]$ must be a linear function of $t$ plus a quadratic polynomial of $x$, under some growth assumptions on $u$.

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arXiv2022

Uniqueness of positive solutions to the higher order Brezis-Nirenberg problem

In this paper, we study the higher order Brezis-Nirenberg problem under the Navier boundary condition \be\label{eq} \begin{cases} (-Δ)^m u=\varepsilon u+u^{p} & \text { in }\, Ω, \\ u>0 & \text { in }\, Ω, \\ u=-Δu=\cdots=(-Δ)^{m-1} u=0 & \text { on }\, \partial Ω, \end{cases} \ee where $Ω$ is a strictly convex smooth bounded domain in $\mathbb{R}^n$ with $n \geq 4m$, $m \in \mathbb{N}_{+}$, $\varepsilon\in (0,λ_{1})$, $λ_{1}$ is the first Navier eigenvalue for $(-Δ)^{m}$ in $Ω$, and $p=\frac{n+2m}{n-2m}$. We prove that the solutions of \eqref{eq} are unique if either $\varepsilon$ close to $λ_1$ or $\varepsilon$ close to 0 and $Ω$ satisfies some symmetry assumptions. The proof is mainly based on our previous works about the blow up analysis and compactness result for solutions to higher order critical elliptic equations and the asymptotic behavior of solutions to \eqref{eq}.

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arXiv2022

Unified results of compactness and existence for prescribing fractional $Q$-curvatures problem

In this paper we study the problem of prescribing fractional $Q$-curvature of order $2σ$ for a conformal metric on the standard sphere $\Sn$ with $σ\in (0,n/2)$ and $n\geq2$. Compactness and existence results are obtained in terms of the flatness order $β$ of the prescribed curvature function $K$. Making use of integral representations and perturbation result, we develop a unified approach to obtain these results when $β\in [n-2σ,n)$ for all $σ\in (0,n/2)$. This work generalizes the corresponding results of Jin-Li-Xiong [Math. Ann. 369: 109--151, 2017] for $β\in (n-2σ,n)$.

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arXiv2022

A weighted Sobolev-Poincaré type trace inequality on Riemannian manifolds

Given $(M, g)$ a smooth compact $(n+1)$-dimensional Riemannian manifold with boundary $\partial M$. Let $ρ$ be a defining function of $M$ and $σ\in(0,1)$. In this paper we study a weighted Sobolev-Poincaré type trace inequality corresponding to the embedding of $W^{1,2}(ρ^{1-2 σ}, M) \hookrightarrow L^{p}(\partial M)$, where $p=\frac{2 n}{n-2 σ}$. More precisely, under some assumptions on the manifold, we prove that there exists a constant $B>0$ such that, for all $u \in W^{1,2}(ρ^{1-2σ}, M)$, $$ \Big(\int_{\partial M}|u|^{p} \,\ud s_{g}\Big)^{2/p} \leq μ^{-1} \int_{M} ρ^{1-2 σ}|\nabla_{g} u|^{2} \,\ud v_{g}+B \Big|\int_{\partial M} |u|^{p-2}u \,\ud s_{g}\Big|^{2/(p-1)}. $$ This inequality is sharp in the sense that $μ^{-1}$ cannot be replaced by any smaller constant. Moreover, unlike the classical Sobolev inequality, $μ^{-1}$ does not depend on $n$ and $σ$ only, but depends on the manifold.

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arXiv2022

Compactness and existence results of the prescribing fractional $Q$-curvatures problem on $\mathbb{S}^n$

This paper is devoted to establishing the compactness and existence results of the solutions to the prescribing fractional $Q$-curvatures problem of order $2σ$ on $n$-dimensional standard sphere when $ n-2σ=2$, $σ=1+m/2,$ $m\in \mathbb{N}_{+}.$ The compactness results are novel and optimal. In addition, we prove a degree-counting formula of all solutions to achieve the existence. From our results, we can know where blow up occur. Furthermore, the sequence of solutions that blow up precisely at any finite distinct location can be constructed. It is worth noting that our results include the case of multiple harmonic.

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