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This paper deals with the quasilinear attraction-repulsion chemotaxis system\begin{align*} \begin{cases}  u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u  -\chi u(u+1)^{p-2}\nabla v  +\xi u(u+1)^{q-2}\nabla w\big)  +f(u),  \\[1.05mm]  0=\Delta v+\alpha u-\beta v,  \\[1.05mm]  0=\Delta w+\gamma u-\delta w \end{cases} \end{align*} in a bounded domain$\Omega \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) with smooth boundary$\partial\Omega$, where $m, p, q \in \mathbb{R}$, $\chi, \xi, \alpha, \beta,\gamma, \delta>0$ are constants. Moreover, it is supposed that the function $f$satisfies $f(u)\equiv0$ in the study of boundedness, whereas, when consideringblow-up, it is assumed that $m>0$ and $f$ is a function of logistic type suchas $f(u)=\lambda u-\mu u^{\kappa}$ with $\lambda \ge 0$, $\mu>0$ and $\kappa>1$sufficiently close to~$1$, in the radially symmetric setting. In the case that$\xi=0$ and $f(u) \equiv 0$, global existence and boundedness have been provedunder the condition $p<m+\frac2n$. Also, in the case that $m=1$, $p=q=2$ and$f$ is a function of logistic type, finite-time blow-up has been established byassuming $\chi\alpha-\xi\gamma>0$. This paper classifies boundedness andblow-up into the cases $p<q$ and $p>q$ without any condition for the sign of$\chi\alpha-\xi\gamma$ and the case $p=q$ with $\chi\alpha-\xi\gamma<0$ or$\chi\alpha-\xi\gamma>0$.

Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic-elliptic attraction-repulsion chemotaxis system