An application of Phragmén–Lindelöf theorem to the existence of ground state solutions for the generalized Schrödinger equation with optimal control

In this paper, we develop optimal Phragmén–Lindelöf methods, based on the use of maximum modulus of optimal value of a parameter in a Schrödinger functional, by applying the Phragmén–Lindelöf theorem for a second-order boundary value problems with respect to the Schrödinger operator. Using it, it is possible to find the existence of ground state solutions of the generalized Schrödinger equation with optimal control. In spite of the fact that the equation of this type can exhibit non-uniqueness of weak solutions, we prove that the corresponding Phragmén–Lindelöf method, under suitable assumptions on control conditions of the nonlinear term, is well-posed and admits a nonempty set of solutions.