Non-degeneracy of solution for critical Lane-Emden systems with linear perturbation

TL;DR

This paper investigates the non-degeneracy of blowing-up solutions for a critical Lane-Emden elliptic system with linear perturbations, providing new methods applicable to related Hamiltonian systems.

Contribution

It proves the non-degeneracy of solutions for a perturbed critical Lane-Emden system, extending previous constructions and introducing new analytical techniques.

Findings

Revisited and analyzed blowing-up solutions from prior work.

Proved the non-degeneracy of these solutions.

Developed new ideas and computational techniques for critical Hamiltonian systems.

Abstract

In this paper, we consider the following elliptic system \begin{equation*} \begin{cases} -\Delta u = |v|^{p-1}v +\epsilon(\alpha u + \beta_1 v), &\hbox{ in }\Omega, \\-\Delta v = |u|^{q-1}u+\epsilon(\beta_2 u +\alpha v), &\hbox{ in }\Omega, \\u=v=0,&\hbox{ on }\partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 3$, $\epsilon$ is a small parameter, $\alpha$, $ \beta_1$ and $ \beta_2$ are real numbers, $(p,q)$ is a pair of positive numbers lying on the critical hyperbola \begin{equation*} \begin{split} \frac{1}{p+1}+\frac{1}{q+1} =\frac{N-2}{N}. \end{split} \end{equation*} We first revisited the blowing-up solutions constructed in \cite{Kim-Pis} and then we proved its non-degeneracy. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems involving critical Halmitonian system and the construction of new solutions.