For $k = 2$ and coefficient functions $p(x)$, $w(x)$, the critical points of the variational optimization problem in $y(x)$
\begin{equation} \text{minimize} \quad \int_a^b p (y')^k dx \qquad \text{subject to} \quad \int_a^b w y^2 dx = 1 \tag{1} \end{equation}
are given by solutions of the Sturm-Liouville problem on an interval $J=(a,b)$
\begin{equation} \tag{2} \left(\frac{k}{2} py'^{(k-1)} \right)' = -\lambda w y \end{equation} with the natural boundary condition \begin{equation} \tag{3} p y'^{(k-1)} = 0 \quad \text{at} \quad a, b \end{equation}
(e.g. Courant, Hilbert - Methods of Mathematical Physics)
I am interested in the case $k=4$ for Neumann Boundary conditions. That is, instead of minimizing the energy $(y')^2$, I want to minimize $(y')^4$ which by my intuition is also a kind of energy minimization, just measured at a higher degree. So, I would expect the solutions to possess many of the qualities known from solutions of Sturm-Liouville problems. In particular I am interested in the Oscillation property and the Spectral Theorem. My questions:
How is that kind of ODE called, what references are there, where that type of equation is studied?
Do solutions obey the Spectral Theorem, i.e. form an orthogonal basis and have ordered eigenvalues?
Do solutions obey the Oscillation property? ($i$th solution has $i$ zeros on $J$) Does at least the first non-constant solution have exctly one zero?