Let $\Omega$ denote a an exterior domain in $R^N$ with smooth boundary.
I am interested in Liouville Theorems related to smooth solutions of
$$\Delta \phi(x) + \gamma \sum_{i,j=1}^N \frac{x_i x_j}{|x|^2} \phi_{x_i x_j}(x) =0$$ in $\Omega$ with $ \phi=0$ on $ \partial \Omega$ and under the assumption that $ \sup_\Omega |x|^{\sigma+1} | \nabla \phi(x)| \le C$. The domain does not contain the origin in its closure.
In the case of $\Omega$ the compliment of the unit ball centred at the origin I can get exact (and optimal) conditions on $\gamma$ and $\sigma$ that give me a Liouville theorem (one writes $ \phi$ with spherical harmonics and then examines the ODE's).
In the case of $\gamma=0$ but a general domain I work on $\Omega_R:=\{x \in \Omega: |x|<R\}$ and multiply $-\Delta \phi$ by $ \phi$ and integrate by parts and then use the bound on $\phi$ to control the boundary terms on $B_R$. This gives an optimal result.
In the case of $ \gamma>0$ and general domain I can try and copy the same approach but it appears either I won't get anything, or at least I won't get optimal results.
QUESTION. Is there a general approach to trying to prove these things? If I let $$ \psi(r) = \int_{|\theta|=1} \phi(r \theta) d \theta$$ for $ r>0$ big enough so $ \partial B_r \subset \Omega$ I see (under the expected assumptions on $\gamma$ and $ \sigma$) that $ \psi(r)$ is constant. Of course since the domain isn't radial its hard to use this for much.
Is there any hope of using a continuity argument to connect the general case to the radial case?
thanks for all comments.
