On the existence of bounded solutions for a nonlinear elliptic system

TL;DR

This paper establishes conditions under which solutions to a nonlinear elliptic system are bounded and proves these conditions are sharp, extending previous results for the case m=1 to higher orders m≥1.

Contribution

It generalizes existing results on boundedness of solutions from the case m=1 to higher-order elliptic systems with Dirichlet boundary conditions.

Findings

Solutions are bounded under specific parameter conditions.

Conditions for boundedness are proven to be sharp.

Results extend previous work for m=1 to all m≥1.

Abstract

This work deals with the system $(-\Delta)^m u= a(x) v^p$, $(-\Delta)^m v=b(x) u^q$ with Dirichlet boundary condition in a domain $\Omega\subset\RR^n$, where $\Omega$ is a ball if $n\ge 3$ 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 \cite{PS}. Our paper generalize to $m\ge 1$ the results of that paper.