Problem on multiple eigenvalues and positive eigenfunctions for a one-dimensional second-order quasilinear equation (Q1930796)

The one-dimensional quasilinear eigenvalue problem \[ u^{\prime \prime} + \lambda f(u) = 0, \;u(x) > 0,\quad x \in (0, l) \] with the boundary conditions \[ u'(0) = 0,\;u(l) = 0 \] is considered, where \(f(u)\) is a locally Lipschitz-continuous function on \([0, \infty)\) and \(f(u) > 0\) if \(u > 0\). Through an expression for \(x(u)\), which is the inverse of the eigenfunction \(u(x)\), the authors analyse a number of versions of the function \(f\) specifying the nonlinearity for which the problem has multiple eigenvalues.