Please help me to solve the attached problem
Existence and uniqueness look like they should be rather standard. You can't solve the problem in full (i.e., exhibit an explicit solution) until you know all the particulars about the boundary, the constants, the RHS's of the PDE's, etc. As for a physics application, you've got yourself a non-homogeneous Laplace equation. That could describe any number of physical situations from electrodynamics to thermal physics.