On the existence of bounded solutions for a nonlinear elliptic system

2010/11/01 by Ricardo G. Duran, Ricardo G. Durán, Duran, Ricardo G. +4
Computer Science · Mathematics · #35J48 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #math.AP #msc:35J48

paper · pdf · doi:10.48550/arxiv.1011.0412

arxiv created 2010/11/01 · openalex publication_date 2010/11/01 · arxiv updated 2010/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This work deals with the system (-Δ)^m u= a(x) v^p, (-Δ)^m v=b(x) u^q with Dirichlet boundary condition in a domain Ω⊂\RRⁿ, 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 \citePS. Our paper generalize to m≥ 1 the results of that paper.

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