Amplitude dependence of a nonlinear eigenvalue correction ($\nu (C)=C^{p-1}\nu (1)$)

Suppose a real simple leading eigenfunction is $y_0=C\varphi$ and a first-order correction satisfies $L_{0}v=-\nu w{y}_0-y_0^p$, where $p\ge 2$ is an integer, $L_0$ is self-adjoint with null space spanned by $\varphi$, the boundary conditions are homogeneous and independent of the parameter, and $w>0$. The Fredholm solvability condition givesIndeed, orthogonality of the forcing to $\varphi$ gives $\nu C\int w{\varphi }^2+C^p\int {\varphi }^{p+1}=0$, and $C\ne 0$ permits division. Thus the numerical eigenvalue correction depends on the chosen leading amplitude; fixing an eigenfunction normalization is part of defining the nonlinear branch. This calculation assumes a perturbative branch and does not by itself establish that branch's existence.