arXiv subjects

Congzheng Xuanyuan

Publications and source records attributed to Congzheng Xuanyuan.

2 recordsLinked to original sources

arXiv2026

Infinitely many sign-changing solutions for critical Hamiltonian systems with linear perturbation

In this paper, we study the following elliptic system \begin{equation}\label{main_1} \begin{cases} -Δu = |v|^{p-1} v + ε(αu + β_1 v), & \text{in } Ω, \\ -Δv = |u|^{q-1} u + ε(β_2 u + αv), & \text{in } Ω, \\ u = v = 0, & \text{on } \partial Ω, \end{cases} \tag{*} \end{equation} where $Ω$ is the unit ball in $\mathbb{R}^N$, $ε$ is a small parameter, $α$, $β_1$ and $β_2$ are real numbers, $(p, q)$ is a pair of positive numbers lying on the critical hyperbola \begin{equation} \frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2}{N}.\nonumber \end{equation} Under suitable assumptions and suitable restrictions on $(p,q)$ and $N$, we construct infinitely many sign-changing solutions to \eqref{main_1} which look like a positive radial solution to \eqref{main_1} crowned by $k$ negative bubbles arranged on a regular polygon of a suitable radius, whose energy can be arbitrarily large.

math.AP↗

arXiv2025

Construction of solutions for a critical elliptic system of Hamiltonian type

We consider the following nonlinear elliptic system of Hamiltonian type with critical exponents: \begin{equation*} \begin{cases} -Δu + V(|y'|,y'')\, u = |v|^{p-1}v, & \text{in } \mathbb{R}^N,\newline -Δv + V(|y'|,y'')\, v = |u|^{q-1}u, & \text{in } \mathbb{R}^N, \end{cases} \end{equation*} where $(y', y'') \in \mathbb{R}^2 \times \mathbb{R}^{N-2}$, $V(|y'|, y'') \not\equiv 0$ is a bounded, nonnegative function on $\mathbb{R}_+ \times \mathbb{R}^{N-2}$ and $p, q > 1$ lie on the critical hyperbola: \[ \frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2}{N}. \] By applying the finite-dimensional reduction method and local Pohozaev identities combined with the Green representation formula and technical analysis, we show that, under the assumptions that $N \ge 5$, $(p,q)$ lies in a certain admissible range, and $r^2 V(r, y'')$ has a stable critical point, the above problem admits infinitely many solutions whose energy can be made arbitrarily large.

math.AP↗