Let (M, g) be a smooth compact Riemannian manifold of dimension \(n\ge 3\). Denote by \(P_{g}\) the GJMS (Graham–Jenne–Mason–Sparling) operator. In this paper, we introduce the GJMS invariant \(\mu \) and we define the first GJMS invariant \(\mu _1\) as the infimum of the first eigenvalue of \(P_{g}\) over the metrics conformal to g and of volume 1. We study when it is attained and whether is equal to \(\mu \) . As an application, we show that the nonlinear GJMS equation \(P_{g}v =\mu _{1}|v|^{N-2}v\) has nodal (sign-changing) solution. When g is Einstein, the above equation has positive solutions if the scalar curvature \(S_{g}>0\).

令（M，  g）为维\（n \ ge 3 \）的光滑紧黎曼流形。用\（P_ {g} \）表示GJMS（Graham–Jenne–Mason–Sparling）运算符。在本文中，我们介绍了GJMS不变式\（\ mu \），并将第一个GJMS不变式\（\ mu _1 \）定义为\（P_ {g} \）的第一特征值在与g和体积1。我们研究何时达到该值以及是否等于\（\ mu \）。作为应用，我们证明了非线性GJMS方程\（P_ {g} v = \ mu _ {1} | v | ^ {N-2} v \）具有节点（符号转换）解。当g如果是爱因斯坦，则如果标量曲率\（S_ {g}> 0 \），则上述方程具有正解。