Eigenvalue analysis for the \(p\)-Laplacian under convective perturbation
Considered is the Dirichlet eigenvalue problem for the one-dimensional \(p\)-Laplacian in the unit interval when the equation is perturbed by convection terms, namely \[ -(\psi_p(u'))'- c\psi_p(u')= \lambda\psi_p(u),\quad 0<x<1,\quad u(0)= u(1)= 0 \] where, for \(p>1\), \(\psi_p(z):=|z|^{p-2}z\) stands for the odd extension of \(z^{p-1}\) while \(c\) can be assumed positive. Using a global phase space analysis the authors establish among others the following results: The problem admits an unbounded sequence of eigenvalues \(\lambda_n= \lambda_n(c)\) such that \((c/p)^p< \lambda_1<\cdots< \lambda_n<\cdots\). Each \(\lambda_n\) is simple. The corresponding eigenfunctions \(u_n\) exhibit \(n-1\) equally spaced zeros in the interval \(0<x<1\). \(\lambda_n= \lambda_n(c, p)\) is a smooth function of \(c\) and \(p\). Asymptotic estimates (as \(c\to\infty\)) are given.
scientific article; zbMATH DE number 1367868
scientific article