That is a series of ODE. You will not be able to solve it with solve(). You could try it with
dsolve([eqn1, eqn2, eqn3, eqn4])
but the reality is that you will get back emptiness.
There is a trivial solution with u, v, r, and p all being constant 0s. Other than that there does not appear to be any closed form solutions.
Any time you have an unknown function involved U(r), V(r), W(r) then you are not going to get a closed form solution except possibly trivial solutions (like the constant zeros.)