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][1]):

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

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 $p_1:\mathbb{R}\rightarrow (0,\infty)$ is sufficiently smooth and integrable (in particular, I'm interested in the case $p_1(x)\propto e^{-x^2}$, but generality is always welcome.)

 1. How can we show that there is no nontrivial solution to (1) in the class of integrable functions on $\mathbb{R}\times[0,1]$?
 2. How can we show that there is a weak/generalized solution consisting of the linear combination of Dirac measures at 0 and 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!

  [1]: https://math.stackexchange.com/questions/2554069/convergence-stability-of-sde-that-depends-on-an-ergodic-process/2555396#2555396