arXiv subjects

Zhengni Hu

Publications and source records attributed to Zhengni Hu.

8 recordsLinked to original sources

arXiv2026

Blow-up solutions for mean field equations with non-quantized singularities on Riemann surfaces with boundary

We study mean field equations with singular sources on a compact Riemann surface with boundary $(Σ,g)$, subject to homogeneous Neumann boundary conditions: \[ -Δ_g v = ρ\left( \frac{V e^{v}}{\int_ΣV e^{v}\, d v_g} - \frac{1}{|Σ|_g}\right) - \sum_{ξ\in Q} \frac{\varrho(ξ)}{2}γ(ξ) \left(δ_ξ- \dfrac{1}{|Σ|_g}\right) \text{in }Σ; \qquad \partial_{ν_g} v = 0 \text{ on }\partialΣ. \] Here, $V$ is a smooth positive function, $ρ$ is a non-negative parameter, $Q\subsetΣ$ is a finite set of prescribed singular points, and the singular weights satisfy $γ(ξ)\in(-1,+\infty)\setminus(\mathbb{N}\cup\{0\})$. The coefficients are given by $\varrho(ξ)=8π$ for $ξ\inΣ\setminus\partialΣ$ and $\varrho(ξ)=4π$ for $ξ\in\partialΣ$. We construct blow-up solutions in the non-quantized singular regime, including purely singular and mixed singular-regular blow-up cases, with parameters approaching resonant values. The construction is achieved via a Lyapunov-Schmidt reduction under suitable stability assumptions. Key words: Singular mean field equations, Blow-up phenomena, Lyapunov-Schmidt reduction, Riemann surfaces with boundary

math.AP↗

arXiv2026

Blow-up Solutions for General Toda Systems on Riemann Surfaces

In this paper, we study general Toda systems with homogeneous Neumann boundary conditions on Riemann surfaces. Assuming the surface satisfies the ``$k$-symmetric'' condition, we construct a family of bubbling solutions using singular perturbation methods, where the concentration rates of different components occur in distinct orders. In particular, we establish the existence of asymmetric blow-up solutions for the $SU(3)$ Toda system. Furthermore, the blow-up points are precisely located at the ``$k$-symmetric'' centers of the surface. Keywords: Toda system, Neumann boundary condition, Blow-up solutions, $k$-symmetry, Finite-dimensional reduction

math.AP↗

arXiv2025

A degree-counting formula for a Keller-Segel equation on a surface with boundary

In this paper, we consider the following Keller-Segel equation on a compact Riemann surface $(Σ, g)$ with smooth boundary $\partialΣ$: \[ -Δ_g u = ρ\Big(\frac{V e^u}{\int_Σ V e^u \mathrm{d} v_g} - \frac{1}{|Σ|_g}\Big) \text{ in } Σ, \quad \text{ with } \partial_{ν_g} u = 0 \text{ on } \partial Σ, \] where $V$ is a smooth positive function on $Σ$ and $ρ> 0$ is a parameter. We perform a refined blow-up analysis of bubbling solutions and establish sharper a priori estimates around their concentration points. We then compute the Morse index of these solutions and use it to derive a counting formula for the Leray-Schauder degree in the non-resonant case (i.e., $ρ\notin 4 π\mathbb{N}$). Our approach follows the strategy suggested by Y. Y. Li [33] and later implemented by C.-S. Lin and C.-C. Chen [15,16] for the mean field equations on closed surfaces and employs techniques from Bahri's critical points at infinity [8].

math.AP↗

arXiv2025

On Solutions for Singular Toda System on Riemann Surfaces with Boundary

This paper studies solutions to a singular $SU(3)$ Toda system with linear source terms on a compact Riemann surface $Σ$ with smooth boundaries $\partialΣ$. We establish the existence of solutions when the parameters are not critical, assuming that Euler characteristic $χ(Σ)<1$ via analyzing the sublevels. Furthermore, we find a sufficient condition that ensures multiple solutions for generic potentials by Morse inequalities and a transversality theorem.

math.AP↗

arXiv2025

Blow-up solutions for the steady state of the Keller-Segel system on Riemann surfaces

We study the following Neumann boundary problem related to the stationary solutions of the Keller-Segel system, a basic model of chemotaxis phenomena: \[ \left\{\begin{array}{ll} -Δ_g u +βu =λ\left(\frac{Ve^u}{\int_Σ Ve^u d v_g}-1\right), &\text { in } \mathringΣ\\ \partial_{ ν_g} u=0, &\text { on } \partial Σ\end{array} \right.,\] on a compact Riemann surface $(Σ, g)$ of unit area, with interior $\mathringΣ$ and smooth boundary $\partial Σ$. Here, $Δ_g$ denote the Laplace-Beltrami operator, $dv_g$ the area element of $(Σ, g)$, and $ν_g$ the unit outward normal to $\partial Σ$ and $λ$ and $β$ are non-negative parameters, $V$ is non-negative with finite zero set. For any integers $m>0$ and $k,l\geq 0$ with $m=2k+l$, we establish a sufficient condition on $V$ for the existence of a sequence of blow-up solutions as $λ$ approaches the critical values $4πm$, which blows up at $k$ points in the interior and $l$ points on the boundary. Moreover, the study expands to the corresponding singular problem.

math.AP↗

arXiv2025

Blow-up solutions for mean field equations with Neumann boundary conditions on Riemann surfaces

On a compact Riemann surface $(Σ, g)$ with a smooth boundary $\partial Σ$, we consider the following mean field equations with Neumann boundary conditions: $$ -Δ_g u = λ\left(\frac{Ve^u}{\int_Σ Ve^u \, dv_g} - \frac{1}{|Σ|_g}\right) \text{ in } Σ\text{ with } \partial_{ν_g} u = 0 \text{ on } \partial Σ, $$ We find conditions on the potential function $V: Σ\to \mathbb{R}^+$ such that solutions exist for the parameter $λ$ when it is in a small right (or left) neighborhood of a critical value $4π(m+k)$ for $k \leq m \in \mathbb{N}_+$ and blow up as $λ$ approaches the critical parameter. The blow-up occurs exactly at $k$ points in the interior of $Σ$ and $(m-k)$ points on the boundary $\partial Σ$.

math.AP↗

arXiv2024

Partial Blow-up Phenomena in the $SU(3)$ Toda System on Riemann Surfaces

This work studies the partial blow-up phenomena for the $SU(3)$ Toda system on compact Riemann surfaces with smooth boundary. We consider the following coupled Liouville system with Neumann boundary conditions: $$ -Δ_g u_1 = 2ρ_1\left( \frac{V_1 e^{u_1}}{\int_Σ V_1 e^{u_1} \, dv_g} - \frac 1 {|Σ|_g}\right) - ρ_2\left( \frac{V_2 e^{u_2}}{\int_Σ V_2 e^{u_2} \, dv_g} - \frac{1}{|Σ|_g}\right) \text{in} \,\mathringΣ$$ and $$ -Δ_g u_2 = 2ρ_2\left( \frac{V_2 e^{u_2}}{\int_Σ V_2 e^{u_2} \, dv_g} - \frac{1}{|Σ|_g}\right) - ρ_1\left( \frac{V_1 e^{u_1}}{\int_Σ V_1 e^{u_1} \, dv_g} - \frac{1}{|Σ|_g}\right) \text{in} \,\mathringΣ$$ with boundary conditions $ \partial_{ν_g} u_1 = \partial_{ν_g} u_2 = 0 \text{ on} \, \partial Σ,$ where $(Σ, g)$ is a compact Riemann surface with the interior $\mathringΣ$ and smooth boundary $\partialΣ$, $ρ_i$ is a non-negative parameter and $V_i$ is a smooth positive function for $i=1,2$. We construct a family of blow-up solutions via the Lyapunov-Schmidt reduction and variational methods, wherein one component remains uniformly bounded from above, while the other exhibits partial blow-ups at a prescribed number of points, both in the interior and on the boundary. This construction is based on the existence of a non-degeneracy solution of a so-called shadow system. Moreover, we establish the existence of partial blow-up solutions in three cases: (i) for any $ρ_2>0$ sufficiently small; (ii) for generic $V_1, V_2$ and any $ρ_2\in (0,2π)$; (iii) for generic $V_1, V_2$, the Euler characteristic $χ(Σ)<1$ and any $ρ_2\in (2π,+\infty)\setminus 2π\mathbb{N}_+$.

math.AP↗

arXiv2024

The Morse property of limit functions appearing in mean field equations on surfaces with boundary

In this paper we study the Morse property for functions related to limit functions of mean field equations on a smooth, compact surface $Σ$ with boundary $\partialΣ$. Given a Riemannian metric $g$ on $Σ$ we consider functions of the form \[ f_g(x) := \sum_{i=1}^mσ_i^2R^g(x_i)+\sum_{i,j=1\ı\ne j}^mσ_iσ_jG^g(x_i,x_j)+h(x_1,\ldots,x_m), \] where $σ_i \neq 0$ for $i=1,\ldots,m$, $G^g$ is the Green function of the Laplace-Beltrami operator on $(Σ,g)$ with Neumann boundary conditions, $R^g$ is the corresponding Robin function, and $h \in \mathcal{C}^{2}(Σ^m,\mathbb{R})$ is arbitrary. We prove that for any Riemannian metric $g$, there exists a metric $\widetilde g$ which is arbitrarily close to $g$ and in the conformal class of $g$ such that $f_{\widetilde g}$ is a Morse function. Furthermore we show that, if all $σ_i>0$, then the set of Riemannian metrics for which $f_g$ is a Morse function is open and dense in the set of all Riemannian metrics.

math.DG↗