Global Weak Solution in a p-Laplacian Attraction-Repulsion Chemotaxis System with Nonlinear Sensitivity and Signal Production

We consider the following p-Laplacian chemotaxis system with logistic source and nonlinear production: $$\{ {array}{ll} uₜ=·(|∇ u|ᵖ⁻²∇ u)-χ ∇·(uα ∇ v)+ξ ∇·(uβ ∇ w)+f(u),\;\;x∈ Ω,\;t>0,\\ vₜ= v-σ v+η u^{k₁},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x∈ Ω,\;t>0,\\ 0= w-δ w+γ u^{k₂},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x∈ Ω,\;t>0, {array}. $$ where $ χ, ξ, α,β,η,σ,γ,δ, k₁, k₂ >0, p≥ 2$, $ Ω ⊂ Rⁿ(n≥2) $ is a smoothly bounded domain. The logistic-type source $f(s)≤κ-μ sᵐ$ for $κ∈ R, μ>0$ and $ m>1$. We obtain the global boundedness of solutions if (i) $m>max\{2k₁,pα/p-1-1\} $, or (ii) $k₂>max\{2k₁-β,pα/p-1-β-1\}$ with $m>max\{2,2α,2β\}$.