Asymptotic behavior of least energy solutions to the Lane-Emden system near the critical hyperbola

Abstract

The Lane-Emden system is written as \begin{equation*} \begin{cases} -Δu = v^p &\text{in } Ω,\\ -Δv = u^q &\text{in } Ω,\\ u, v > 0 &\text{in } Ω,\\ u = v = 0 &\text{on } \partial Ω\end{cases} \end{equation*} where $\Omega$ is a smooth bounded domain in the Euclidean space ${\mathbb{R}}^n$ for $n\ge 3$ and $0<p<q<\infty$. The asymptotic behavior of least energy solutions near the critical hyperbola was studied by Guerra \cite{G} when $p\ge 1$ and the domain is convex. In this paper, we cover all the remaining cases $p<1$ and extend the results to any smooth bounded domain.

45 pages