Fix boundary condition at infinity

Question

I am working with this complicated PDE:

ClearAll["Global`*"]
m = 1;
r[u_, v_] := 2 m + 2 m ProductLog[-((u v)/E)];
F[u_, v_] := (32 m^3)/r[u, v] Exp[-(r[u, v]/(2 m))];
P1[u_, v_, l_] = (D[F[u, v], u] D[F[u, v], v])/F[u, v]^2 + (
   D[r[u, v], v] D[F[u, v], u])/(r[u, v] F[u, v]) - (
   D[r[u, v], u] D[r[u, v], v])/r[u, v]^2 - D[F[u, v], u, v]/
   F[u, v] - 3 D[r[u, v], u, v]/r[u, v] - 
   F[u, v]/(4 r[u, v]^2) l (l + 1);
Q1[u_, v_, l_] = -(F[u, v]/(4 r[u, v]^2)) l (l + 1) + (
   D[F[u, v], u] D[F[u, v], v])/F[u, v]^2 + (
   D[F[u, v], v] D[r[u, v], u])/(F[u, v] r[u, v]) - (
   D[r[u, v], u] D[r[u, v], v])/r[u, v]^2 - D[F[u, v], u, v]/
   F[u, v] - 3 D[r[u, v], u, v]/r[u, v];
P2[u_, v_] = -((D[F[u, v], u] D[r[u, v], v])/(F[u, v] r[u, v])) + 
   D[r[u, v], u]^2/r[u, v]^2 + D[r[u, v], {u, 2}]/r[u, v];
Q2[u_, v_] = -((D[F[u, v], v] D[r[u, v], u])/(F[u, v] r[u, v])) + 
   D[r[u, v], v]^2/r[u, v]^2 + D[r[u, v], {v, 2}]/r[u, v];
V[u_, v_] = (Q2[u, v] - P2[u, v])/2 + 
   1/2 ((D[F[u, v], u] D[r[u, v], v] - D[F[u, v], v] D[r[u, v], u])/(
     F[u, v] r[u, v]));
W[u_, v_, 
   l_] = ((P1[u, v, l] + Q1[u, v, l])/2 + (Q2[u, v] + P2[u, v])/
     2) + (-3/4 (D[F[u, v], v] D[F[u, v], u])/F[u, v]^2 - 
     1/2 (D[r[u, v], u] D[F[u, v], v])/(r[u, v] F[u, v]) - 
     1/2 (D[r[u, v], v] D[F[u, v], u])/(r[u, v] F[u, v]) + 
     2 (D[r[u, v], u] D[r[u, v], v])/r[u, v]^2 + 
     1/2 D[F[u, v], u, v]/F[u, v] + D[r[u, v], u, v]/r[u, v]);
sol = NDSolve[{D[f[u, v], u, 
     v] == -(W[u, v, 2]) f[u, v] - (V[u, v]) g[u, 
       v] - (+(1/2) D[g[u, v], u] (D[F[u, v], v]/F[u, v]) - 
       1/2 D[g[u, v], v] (D[F[u, v], u]/F[u, v])), 
   D[g[u, v], u, 
     v] == -(W[u, v, 2]) g[u, v] - (V[u, v]) f[u, 
       v] - (-(1/2) D[f[u, v], u] (D[F[u, v], v]/F[u, v]) + 
       1/2 D[f[u, v], v] (D[F[u, v], u]/F[u, v])), 
   f[0.001, v] == 10, f[u, -0.001] == 10, 
   g[0.001, v] == -8 , 
   g[u, -0.001] == -8}, {f, g}, {u, 0.001, 
   120.1}, {v, -120.1, -0.001}, PrecisionGoal -> 0, 
  WorkingPrecision -> MachinePrecision, MaxStepSize -> 1]

I know that the solution must be bounded at $v=0=u$ and that it must die off at infinity, so I have imposed some random values at those points, as a solution I have the following plot for $f(u,v)$ and an analogous one for $g(u,v)$. It can be seen that the solution oscillates and decays slowly and that the condition $f[0.001,v]=10$ is not a good choice.

I have tried to use a function such as $\exp(a v)\cos(b v)$, but it made things worse, so I was thinking to enforce the othe condition, i.e. that the solution must die off ant infinity, but I don't know how to impose it. How can I do it?

I am sorry I could not find a MWE.