Consider the problem $$ \left\lbrace \begin{array}[l] --\Delta_n u=f \textrm{ in }\Omega\\ -\Delta_m u=g\textrm{ in }\Omega\\ \textrm{s.t. Dirichlet or Neumann Boundary conditions} \end{array} \right. $$ where $\Delta_n$ denotes the Laplacian operator in $n$ first coordinates of $u:\mathbb{R}^{n+m}\to\mathbb{R}$ and $\Delta_m$ in the last $m$ coordinates. Of course, if we consider the Neumann boundary, one might suppose the necessary conditions on the forcing terms $f,g$.
Do anyone know results concerning the existence and uniqueness of solutions for this kind of problem? Also, how to search this kind of problem?
Thank you all in advance!!