Ground State For The X-Ray Free Electron Laser Schrodinger Equation With Harmonic Potential

We study the following X-ray free electron laser (XFEL) Schrodinger equation with harmonic potential-Delta u+(vertical bar x vertical bar(2)+ beta/vertical bar x vertical bar)u+lambda(vertical bar x vertical bar(-1) *vertical bar u vertical bar(2))u - omega u + vertical bar u vertical bar(p-1)u, x is an element of R-3where p is an element of (1,2) and beta, lambda, omega is an element of R are parameters. By variational methods we prove the existence of ground states and normalized solutions for (0.1) under suitable assumptions on the parameters. In addition, numerical tests are performed to validate the theoretical results. (C) 2021 Elsevier Inc. All rights reserved.