Infinitely Many Normalized Solutions for a Quasilinear Schrödinger Equation

In this paper, we are concerned with the following quasilinear Schrödinger equation \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+ \mu u-\Delta (u^2) u=g(u)~~\hbox {in}~~\mathbb {R}^N,\\ \int _{\mathbb {R}^N}|u|^2dx=m,\\ u\in H^1(\mathbb {R}^N), \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\mathrm{\Delta }u+\mu u-\mathrm{\Delta }(u^2)u=g(u)\text{in}{\mathbb{R}}^N,\hfill \\ {\int }_{{\mathbb{R}}^N}{|u|}^{2}dx=m,\hfill \\ u\in H^1({\mathbb{R}}^N),\hfill \end{array}\right)\end{array}$$ where \(N\ge 2\) $N\ge 2$ , \(m>0\) $m>0$ is a given constant, \(\mu \in \mathbb {R}\) $\mu \in \mathbb{R}$ is a Lagrange multiplier. Under the almost optimal assumptions of g, the existence of infinitely many normalized solutions is obtained via a minimax argument. Moreover, we give a new strategy for finding a minimizer for constraint problems with nonhomogeneous nonlinearities.