On Hamiltonian systems with critical Sobolev exponents

TL;DR

This paper investigates the existence of positive solutions to perturbed critical Lane-Emden systems on bounded domains, revealing new phenomena in the three-dimensional case related to solution existence depending on perturbation types.

Contribution

It extends classical results by solving the problem for all dimensions $N \\geq 4$ and uncovers novel solution existence phenomena in the critical dimension $N=3$.

Findings

Solutions exist for all $N \\geq 4$.

In $N=3$, solution existence depends on the type of perturbation.

New phenomena in the critical hyperbola for $N=3$.

Abstract

In this paper we consider lower order perturbations of the critical Lane-Emden system posed on a bounded smooth domain $\Omega \subset \mathbb{R}^N$, with $N \geq3$, inspired by the classical results of Brezis and Nirenberg \cite{BrezisNirenberg1983}. We solve the problem of finding a positive solution for all dimensions $N \geq 4$. For the critical dimension $N=3$ we show a new phenomenon, not observed for scalar problems. Namely, there are parts on the critical hyperbola where solutions exist for all $1$-homogeneous or subcritical superlinear perturbations and parts where there are no solutions for some of those perturbations.