Existence of global solutions to a quasilinear Schrödinger equation with general nonlinear optimal control conditions

Abstract

Abstract In this article, we study a modified maximum principle approach under condition on the weight of the delay term in the feedback and the weight of the term without delay. On that basis, we prove the existence of global solutions for a quasilinear Schrödinger equation in an unbounded domain with a general nonlinear nonlinear optimal control condition in the weakly nonlinear internal feedback. The equation includes many special cases such as classical Schrödinger equations, fractional Schrödinger equations, and relativistic Schrödinger equations, etc. Our results are established by means of the fixed point theory associated with the Schrödinger operator in suitable b-metric spaces. Moreover, we establish general stability estimates by using some properties of Schrödinger convex functions.

Keywords

• Quasilinear Schrödinger equation
• Global solution
• Maximum principle approach