I am considering the following eigenvalue problem in $\Omega=\mathbb{R}^n_+$ $$-\operatorname{div}(a(x)\nabla \varphi) = \lambda a(x)w(x)\varphi$$ where the weights $a>0$ and $w\in L^{\infty}$ (and is a radial function such that $w>0$), $\lambda>0$ and $\varphi$ lives in the appropriate weighted Sobolev space. I am interested in any results involving the dimension of the second Eigenspace associated with the second smallest eigenvalue to this equation.
I already have shown a weighted Sobolev embedding type estimate, $$\int_{\Omega} a(x) |u|^{2^*}dx \leq C\int_{\Omega} a(x)|\nabla u|^2 dx$$ for functions $u\in C^{\infty}_c(\mathbb{R}).$ The function $w$ is an extremizer for the above inequality and satisfies, $$\int_{\Omega} a(x) |w|^{2^*}dx = C\int_{\Omega} a(x)|\nabla w|^2 dx$$ and $\frac{\partial w}{\partial x_n} =0$ on $\partial \Omega.$
Usually, if $a$ was radial I would use Separation of Variables and Sturm-Liouville Theory to arrive at some conclusion but this will not work here. I also tried looking at some papers related to weighted elliptic eigenvalue problems, but most of the papers were concerned with the first eigenvalue of this equation, so I am not sure if I can use the techniques developed in those articles.
