Let's consider a compact subset $\Omega\subset\mathbb R^n$ without boundary. For example, a unit sphere in $\mathbb{R}^3$. In that case, do we know the existence of solution to a second order elliptic equation: $$Lu = f \mbox{ on } \Omega,$$ where $f$ is a smooth function.
When I tried to find a such theorem in Evan's PDE book, there is a Theorem for weak solution existence when $\Omega$ is a domain and has a boundary condition(like Dirichlet or Neumann boundary condition). However, I cannot find any result regarding a subset without boundary. I don't need to find an exact solution here, just wanted to know the existence of a solution. Could anyone help this problem?
