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" $P,Q,W$.