Global weak solutions for an attraction-repulsion chemotaxis system with p-Laplacian diffusion and logistic source

This paper is concerned with the following attraction-repulsion chemotaxis sys-tem withp-Laplacian diffusion and logistic source: {u(t)=del<middle dot>(|del u|(p-2)del u)-chi del<middle dot>(u del v) +xi del<middle dot>(u del w) +f(u), x is an element of ohm, t >0 , v(t)=4v-beta v+alpha u(k)1, x is an element of ohm, t >0 0 =4w-delta w+gamma u(k)2, x is an element of ohm, t >0, u(x,0) =u(0)(x), v(x,0) =v(0)(x), w(x,0) =w(0)(x), x is an element of ohm. The system here is under a homogenous Neumann boundary condition in a bounded domain ohm subset of Rn(n >= 2), with chi,xi,alpha,beta,gamma,delta,k1,k2>0,p >= 2. In addition, the functionfis smoothand satisfies that(f) (s) <= kappa-mu s(l) for all s >= 0, with kappa is an element of R,mu >0,l >1. It is shown that (i)if (l) >max{2k1,2k1n2+n+1p-1}, then system possesses a global bounded weak solution and (ii)if k(2)>max {2k(1-1),2k1n/2+n+2-p/p-1}withl >2, then system possesses a global bounded weaksolution.