Asymptotic behavior of the value function is studied in an infinite horizon optimal control problem with an unlimited integrand index discounted in the objective functional. Optimal control problems of such type are related to analysis of trends of trajectories in models of economic growth. Stability properties of the value function are expressed in the infinitesimal form. Such representation implies that the value function coincides with the generalized minimax solution of the Hamilton-Jacobi equation. It is shown that that the boundary condition for the value function is substituted by the property of the sublinear asymptotic behavior. An example is given to illustrate construction of the value function as the generalized minimax solution in economic growth models.