The Lane-Emden system near the critical hyperbola on nonconvex domains

TL;DR

This paper investigates the asymptotic behavior of minimal energy solutions to the Lane-Emden system near the critical hyperbola on nonconvex domains, removing the convexity assumption and addressing boundary boundedness challenges.

Contribution

It extends previous results by removing the convexity condition on the domain and establishes uniform boundedness of solutions near the boundary for the system.

Findings

Solutions remain uniformly bounded near the boundary as parameters approach the critical hyperbola.

The removal of convexity assumptions broadens the applicability of the results.

A new contradiction argument using Pohozaev identity is developed for nonconvex domains.

Abstract

In this paper we study the asymptotic behavior of minimal energy solutions to the Lane-Emden system $-\Delta u = v^p$ and $-\Delta v = u^q$ on bounded domains as the index $(p,q)$ approaches to the critical hyperbola from below. Precisely, we remove the convexity assumption on the domain in the result of Guerra \cite{G}. The main task is to get the uniform boundedness of the solutions near the boundary because it is difficulty to adapt the moving plane method for the system on nonconvex domains if $\max \{p,q\} > \frac{n+2}{n-2}$. For the purpose, we shall derive a contradiction by exploiting carefully the Pohozaev type identity if the maximum point approaches to the boundary.