The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post):

$$\mathcal{A}p=0, \quad p\in C^2(\mathbb{R}\times[0,1]) \tag1$$

where $$\mathcal{A}=-x\partial_y[y(1-y)\cdot]+\frac12 x^2\partial^2_y[y^2(1-y)^2\cdot]+x\partial_x+\frac12\partial^2_x+q(x)\partial_x. \tag2 $$

Here, $q(x)=\partial_x\log p_1(x)$, where $p_1(x)\propto e^{-x^2}$.

The role of $p_1(x)$ is that it is the stationary density of the first component of the process. And $p_1(x)p(x,y)$, where $p$ is the solution to the above PDE, is supposed to be the joint stationary density (if a solution exists). Therefore we may ask the following questions:

1. How can we show that there is no nontrivial solution to (1) such that $p_1\cdot p$ is in the class of integrable functions on $\mathbb{R}\times[0,1]$?
2. How can we show that there is a weak/generalized solution to (1) consisting of the linear combination of Dirac measures at $y=0$ and $y=1$?
3. What can be said more generally about the singularity of elliptic PDEs that have zeros in their coefficients? What are necessary and sufficient conditions such that the solution space of such a PDE is spanned by the Dirac measures at the points where the coefficients vanish?

Partial answers are very much welcome!