The existence and multiplicity of L²-normalized solutions to nonlinear Schrödinger equations with variable coefficients

The existence of L² –normalized solutions is studied for the equation −Δu + μu = f (x, u) in R^N, ∫_RN u² dx = m. Here m > 0 and f (x, s) are given, f (x, s) has the L²-subcritical growth and (μ, u) ∈ R × H¹(R^N) are unknown. In this paper, we employ the argument in Hirata and Tanaka (“Nonlinear scalar field equations with L² constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka (“Nonlinear scalar field equations with L² constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry.