Three-term spectral asymptotics for nonlinear Sturm-Liouville problems
Here, the nonlinear Sturm-Liouville problem \[ -u''(t)+|u(t)|^{p-1}u(t)+f(u(t))=\lambda u(t),\quad u(0)=0,\quad u(1)=0, \] is considered on the interval \((0,1)\). The main result deals with a three-term asymptotic formula for the \(n\)th eigencurve \(\lambda_n(\alpha)\) (associated with the eigenfunction \(u_{n,\alpha}\) with \(n-1\) simple interior zeros and \(\|u_{n,\alpha}\|_2=\alpha)\) as \(\alpha\to +\infty\) and for the width of the boundary layer of \(u_{n,\alpha}\) as \(\alpha\to +\infty\). Furthermore, there is also proved that if the function \(f\) is of a special type, the assumptions that the main result holds can be weakened.
scientific article; zbMATH DE number 1836997