Global boundedness of weak solutions for an attraction-repulsion chemotaxis system with p-Laplacian diffusion and nonlinear production

This paper is concerned with the following attraction-repulsion chemotaxis system with p-Laplacian diffusion and nonlinear production: $ u_{t} = \nabla\cdot(|\nabla u|^{p-2}\nabla u)-\chi \nabla\cdot(u \nabla v)+\xi \nabla\cdot(u \nabla w)+f(u) $, $ v_{t} = \triangle v-\beta v+\alpha u^{k_{1}} $, $ w_{t} = \triangle w-\delta w+\gamma u^{k_{2}} $, under homogenous Neumann boundary condition in a bounded domain $ \Omega \subset \mathbb{R}^{n}(n\geq2) $, with $ \chi, \xi, \alpha,\beta,\gamma,\delta, k_{1}, k_{2} >0, p>1 $. In addition, the function $ f $ satisfying $ f(s)\equiv 0 $ or generalizing the logistic-type source $ f(s) = \kappa s-\mu s^{l} $ for all $ s\geq0 $ with $ \kappa\in \mathbb{R}, \mu>0, l>1 $. It is shown that (ⅰ) When $ f(u)\equiv0 $, if $ p>\frac{n(\max\{k_{1},k_{2}\}+2)}{n+1} $ or $ 1<p\leq\frac{n(\max\{k_{1},k_{2}\}+2)}{n+1} $ with $ \|u_{0}\|_{L^{\frac{(\max\{k_{1},k_{2}\}-p+2)n}{p}}(\Omega)} $ is small, the problem possesses a global bounded weak solution. (ⅱ) When $ f(u) = \kappa u-\mu u^{l} $, if $ \max\{k_{1},k_{2}\}< l-1 $ or $ \max\{k_{1},k_{2}\} = l-1 $ with large $ \mu>0 $, the problem possesses a global bounded weak solution.