On the existence of bounded solutions for a nonlinear elliptic system

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This work deals with the system (-Δ)ᵐ u= a(x) vᵖ, (-Δ)ᵐ v=b(x) uq with Dirichlet boundary condition in a domain Ω⊂ⁿ, where Ω is a ball if n≥ 3 or a smooth perturbation of a ball when n=2. We prove that, under appropriate conditions on the parameters (a,b,p,q,m,n), any non-negative solution (u,v) of the system is bounded by a constant independent of (u,v). Moreover, we prove that the conditions are sharp in the sense that, up to some border case, the relation on the parameters are also necessary. The case m=1 was considered by Souplet in PS. Our paper generalize to m≥ 1 the results of that paper.