Consider a PDE system \begin{cases} -\Delta u = \Lambda H(u) & \quad\text{in } \Omega, \\ u = 0 & \quad\text{on } \partial\Omega, \end{cases} where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $\Lambda = \operatorname{diag}(\lambda_1,\ldots,\lambda_m)$ is a given diagonal matrix and $H=(H_1,\ldots,H_m):\mathbb{R}^m\to\mathbb{R}^m$ is a given $C^1$ mapping, $u:\mathbb{R}^n\to\mathbb{R}^m$ is the unknown.
In the paper rigidity results for stable solutions of symmetric systems, the author said that a solution $u$ of the above system is said to be a stable solution if the linearized operator has a positive first eigenvalue.
So what is the linearized operator of the system and what equations does a stable solution satisfy?