# spherical mean of solution of the helmholtz equation

I'm stuck with this problem. Given a domain $\Omega \subset \mathbb{R}^3$ where the function $u$ satisfies: $u_{xx} +u_{yy}+u_{zz} + k^2 u = 0$, I am asked to find the spherical mean over the sphere $\{(x, y, z)\in \mathbb{R}^3; ||(x-x_0, y-y_0, z-z_0)||=R\} \subset \Omega$.

I obviously thought of trying to adapt the mean value property for harmonic functions but to no avail. I then sought to find a general form of the solution using separation of variables. This way I believe that, if using spherical coordinates, the azimuthal factor of the solution is proportional $\Psi(\psi)=e^{in\psi}$, which means that when integrating out the azimuthal angle in $\int u d\sigma=-1/k^2\int \nabla^2 u d\sigma$ I will get zero by periodicity. I feel this solution is probably wrong. Any help would be appreciated. Thanks!