I presume that it means I have to discard two of the three conditions related to u[r,0]=1

still I get an error of the same kind:

  K1 = i;
  V = Tan[x];
  P = r^2/(Cos[y])^5

  eq = K1*r^3*D[u[r, y], {r, 3}] + r^2*D[u[r, y], {r, 2}] + r*K1*D[V^2*u[r, y], {r, 1}] + V*r*K1*D[u[r, y], {r, 1}] + V*r*K1*D[V*u[r,y], {r,1}]+ V^2*r*K1*D[u[r, y], {r, 1}] + 
V^3*K1*D[u[r, y], {r, 1}] == 2*u[r,y]*P

 sol = u[r, y] /. 
 NDSolve[{eq,
 u[0, y] == 1,
 Derivative[1, 0][u][0, y] == 0, 
 Derivative[2, 0][u][0, y] == 10, 
 Derivative[3, 0][u][0, y] == 0, 

 u[r, 0] == 1, 
 Derivative[0, 1][u][r, 0] == 0,
 u, {r, 0, 3}, {y, 0, 3},
 MaxSteps -> Infinity, PrecisionGoal -> 1,
 AccuracyGoal -> 1, 
 Method -> {"MethodOfLines", 
 "SpatialDiscretization" -> {"TensorProductGrid", 
 "MinPoints" -> 32, "MaxPoints" -> 32, "DifferenceOrder" -> 2},
 Method -> {"Adams", "MaxDifferenceOrder" -> 1}}] // 

 Plot3D[sol, {r, 0, 3}, {y, 0, 3}, AxesLabel -> Automatic]

0.00021449999999999998` cannot be used as a variable.

Thank you Gianluca.

I am trying this variant:

  K1 = i;
  V = Tan[x];
  P = r^2/(Cos[y])^5

  eq = K1*r^3*D[u[r, y], {r, 3}] + r^2*D[u[r, y], {r, 2}] + r*K1*D[V^2*u[r, y], {r, 1}] + V*r*K1*D[u[r, y], {r, 1}] + V*r*K1*D[V*u[r,y], {r,1}]+ V^2*r*K1*D[u[r, y], {r, 1}] + 
V^3*K1*D[u[r, y], {r, 1}] == 2*u[r,y]*P

 sol = u[r, y] /. 
 NDSolve[{eq,
 u[0, y] == 1,
 Derivative[1, 0][u][0, y] == 0, 
 Derivative[2, 0][u][0, y] == 10, 
 Derivative[3, 0][u][0, y] == 0, 

 u[r, 0] == 1, 
 Derivative[0, 1][u][r, 0] == 0,
 Derivative[0, 2][u][r, 0] == 10,    
 Derivative[0, 3][u][0, y] == 0}, 
 u, {r, 0, 3}, {y, 0, 3},
 MaxSteps -> Infinity, PrecisionGoal -> 1,
 AccuracyGoal -> 1, 
 Method -> {"MethodOfLines", 
 "SpatialDiscretization" -> {"TensorProductGrid", 
 "MinPoints" -> 32, "MaxPoints" -> 32, "DifferenceOrder" -> 2},
 Method -> {"Adams", "MaxDifferenceOrder" -> 1}}] // 

 Plot3D[sol, {r, 0, 3}, {y, 0, 3}, AxesLabel -> Automatic]

However, I get that

NDSolve::dsvar: 0.00021449999999999998` cannot be used as a variable.