References for "Multidimensional Sturm-Liouville Theory"

Question

I am interested in what I'll call (perhaps erroneously) multivariate Sturm–Liouville theory, i.e., solutions to equations of the form $$\nabla\cdot(P(x)\nabla Y)+Q(x)Y=-\lambda W(x)Y\tag{1}$$ for $\lambda\in\mathbb R$, $Y:\mathbb R^d\to\mathbb R$, and $P,Q,W:\mathbb R^d\to\mathbb R$.

If $d=1$, then, as shown on the Wikipedia page, there is a very well developed theory for solutions of $(1)$ on an interval $[a,b]$ with fixed boundary conditions. This is called Sturm-Liouville theory.

I suspect that such a theory has been generalized to higher dimensions. However, after googling for a while using keywords such as "multivariate Sturm-Liouville theory", I'm starting to suspect that the study of $(1)$ in higher dimensions has a different name, since I've completely failed to find good comprehensive resources on such problems.

Question: Are there textbooks that treat multivariate problems such as $(1)$ in detail? I'm especially interested in fixed point-type arguments for existence and uniqueness, as well as continuity results for the solution $(Y,\lambda)$ with respect to the "data" $P,Q,W$.

Answer 1

Assuming $P$ has constant sign, this is a well-studied class of problems, with the usual terminology being something like "weighted elliptic eigenvalue problem".

In the case $W=1,$ you have the standard eigenvalue problem for the divergence-form linear elliptic operator $L=\operatorname{div} P\, \nabla + Q,$ which is usually analyzed using Fredholm theory - see, for example, the end of Chapter 8 of Elliptic PDE of Second Order by Gilbarg and Trudinger. There has been a huge amount written about elliptic eigenvalue problems which you should have no problems finding online.

When you add the weight $W,$ you're in to marginally less standard territory; but there's still a lot out there. If $W>0$ then I'm guessing you can say quite a bit by rewriting the problem as a standard eigenvalue problem for the unknown $U = WY.$ In more generality, it seems there have been many articles written on the topic of elliptic eigenvalue problems with an indefinite weight function.

Answer 2

Haberman briefly discusses this in Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (Classic Version), 2018, section 7.4. However, he lets $P = W = 1$ and $Q = 0$ before establishing theorems. He calls this the Helmholtz equation and calls the problem (with certain boundary conditions) a "multidimensional eigenvalue problem." Furthermore, he says it is "directly analogous to the one-dimensional regular Sturm–Liouville eigenvalue problem."