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 $\mathrm{\Omega }$ is the unit ball in ${\mathbb{R}}^N$, $\epsilon$ is a small parameter, $\alpha$, ${\beta }_1$ and ${\beta }_2$ 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.