In our research, we focus on the existence, non-existence, and multiplicity of positive solutions to a Quasilinear Schrödinger equation in the form:
\[
-\Delta u + \lambda u + \frac{k}{2}[\Delta(u^2)]u = f(u), \quad u \in H^1(\mathbb{R}^N)
\]
with prescribed mass:
\[
\int_{\mathbb{R}^N} |u|^2 \, dx = c,
\]
where $N \ge 3$. The dual approach is used to transform this equation into a corresponding semi-linear 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 $\lambda \to 0^+$ or $\lambda \to +\infty$, and identifying a continuum of unbounded solutions in $(0, +\infty) \times H^1(\mathbb{R}^N)$.