= Amplitude dependence of a nonlinear eigenvalue correction
{title2=$\nu(C)=C^{p-1}\nu(1)$}
Suppose a real simple leading <eigenfunction> is $y_0=C\phi$ and a first-order correction satisfies $L_0v=-\nu w y_0-y_0^p$, where $p\geq2$ is an <integer>, $L_0$ is <self-adjoint> with <null space> spanned by $\phi$, the <boundary conditions> are homogeneous and independent of the parameter, and $w>0$. The <Fredholm solvability condition> gives
$$
\nu(C)=-C^{p-1}\frac{\int\phi^{p+1}\,dx}{\int w\phi^2\,dx}.
$$
Indeed, <orthogonality> of the forcing to $\phi$ gives $\nu C\int w\phi^2+C^p\int\phi^{p+1}=0$, and $C\ne0$ 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.
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