I need to solve a stationary density distribution of a Fokker Planck equation $-\nabla \cdot(\mu \nabla V) + \frac{1}{2}\sigma^2 \Delta\mu = 0 $ where $\mu$ is the stationary density of the SDE:
$dX = -\nabla Vdt+\sigma dW_t$
where $\mu \ge0, \int_\Omega\mu dx= 1$
I tried to solve this by Finite element method by obtaining the weak formulation: $\frac{1}{2}\sigma^2\int_{\Omega} \nabla \mu \cdot \nabla \phi dx - \int_{\Omega} (\Delta V \mu) \phi dx - \int_{\Omega} (\nabla V \cdot \nabla \mu)\phi dx =0$
Any by supposing the stationary density vanishes at the boundary $\partial \Omega$, one has the Dirichlet boundary condition $\mu=0|\partial\Omega$.
I set $V = \frac{x_1^4}{4}+\frac{x_1^2}{2}+\frac{x_2^2}{2}$ tried to solve the PDE with these conditions on a Disk with radius 5 centred at the origin by Finite element method but the result is identically $0$ for $\mu$ after applying the boundary conditions.
I did some research and from "Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems", "https://link.springer.com/article/10.1007/BF02716786"
one says that "For the stationary problem setting $\dot p = 0$, we get the homogeneous equation $Kp = 0$. The direct solution of the stationary problem is difficult as the system admits both a trivial solution $p = 0$ and a nontrivial solution through enforcement of the normalisation condition.... An additional boundary condition can be applied making the degree of freedom at the origin constrained."
I want to know how may I constraint my PDE/boundary conditions to obtain the non-trivial solution $\mu$ without any other known information about $\mu$?