Existence of solutions of filtration problems with multi-valued law in nonhomogeneous media in the presence of a point source

We formulate a generalized problem of filtration of incompressible fluid governed by a multi-valued law with a linear growth at infinity in nonhomogeneous media in the presence of a point source. We used an additive selection of a feature associated with the singularity of the right side. The solution is represented in the form of the sum of the known solution of a certain linear problem with a point source in the right side, and the unknown term. As for the unknown term, the problem is reduced to the solution of mixed variational inequality in Hilbert space. The existence theorem is proved.