In our research, we focus on the existence, non-existence, and multiplicity of positive solutions to a Quasilinear SchrΓΆdinger equation in the form: ββπ’ + ππ’ + π 2 [β(π’ 2)]π’ = π(π’), π’ β π» 1 (β π΅) With prescribed mass: β« β π |π’| 2 ππ₯ = π, Here π β₯ 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 π(π ) that fall into mass subcritical, critical, or supercritical categories. Key aspects of this study include examining the positive solutions' asymptotic behaviors as π β 0 + ππ π β +β and identifying a continuum of unbounded solutions in (0, +β) Γ π» 1 (β π΅).
