Abstract

In our research, we focus on the existence, non-existence, and multiplicity of positive solutions to a Quasilinear Schrödinger equation in the form: −∆𝑢 + 𝜆𝑢 + 𝑘 2 [∆(𝑢 2)]𝑢 = 𝑓(𝑢), 𝑢 ∈ 𝐻 1 (ℝ 𝑵) With prescribed mass: ∫ ℝ 𝑁 |𝑢| 2 𝑑𝑥 = 𝑐, Here 𝑁 ≥ 3, The dual approach is used to transform this equation into a corresponding semilinear form. Then, we implement a global branch approach, adeptly handling nonlinearities 𝑓(𝑠) that fall into mass subcritical, critical, or supercritical categories. Key aspects of this study include examining the positive solutions' asymptotic behaviors as 𝜆 → 0 + 𝑜𝑟 𝜆 → +∞ and identifying a continuum of unbounded solutions in (0, +∞) × 𝐻 1 (ℝ 𝑵).