In this paper, we study the existence and multiplicity of normalized solutions to the quasilinear Schrödinger equation with prescribed mass where \(N \ge 2, c>0\) is a given constant, \(\lambda \) is a part of the unknown which arises as a Lagrange multiplier. The nonlinearity g is allowed to be strongly sublinear at the origin, i.e., \(\begin{aligned} \lim _{s \rightarrow 0} \frac{g(s)}{s}=-\infty , \end{aligned}\) which includes the logarithmic nonlinearity \(g(s)=s \log s^2\) . The quasilinear term \(\Delta (u^2)u\) and the strongly sublinear term make the associated energy functional not well defined in \(H^1(\mathbb {R}^N)\) . To overcome this difficulty, we use the dual approach and consider a family of approximating problems in \(H^1(\mathbb {R}^N)\) . Then, we prove that such a family of solutions converges to a least-energy solution of the original problem. Additionally, under certain assumptions about g that allow us to work in a suitable subspace of \(H^1(\mathbb {R}^N)\) , we prove the existence of infinitely many solutions to \(\left( \mathcal {P}_\lambda \right) \) .
