This paper is concerned with the quasilinear attraction-repulsion chemotaxis system with flux limitation, \(\begin{aligned} {\left\{ \begin{array}{ll} u_t=\nabla \cdot \left( (u+1)^{m-1}\nabla u -\dfrac{\chi u(u+1)^{p-2}}{(1+|\nabla v|^2)^{k}}\nabla v +\dfrac{\xi u(u+1)^{q-2}}{(1+|\nabla w|^2)^{\ell }}\nabla w\right) , \\ 0=\Delta v+\alpha u^{\lambda }-\beta v, \\ 0=\Delta w+\gamma u^{\theta }-\delta w \end{array}\right. } \end{aligned}\) in an open ball in \(\mathbb {R}^n\) \((n \ge 2)\) , where \(m, p, q \in \mathbb {R}\) , \(k, \ell \ge 0\) , \(0<\lambda \le 1\) , \(0<\theta \le 1\) , \(\chi , \xi , \alpha , \beta , \gamma , \delta >0\) are constants. Boundedness of solutions to this system is known only in the one-dimensional case under some conditions on the parameters and initial data. However, there is no information about the question whether boundedness can be obtained in the higher-dimensional case. The purpose of this paper is to give an answer to this question for all \(n \ge 2\) .
