Existence of infinitely many normalized radial solutions for a class of quasilinear Schrödinger-Poisson equations in ${\mathbb{R}}^3$

In this paper, we study the existence of infinitely many normalized radial solutions for the following quasilinear Schrödinger-Poisson equations:

$$-\mathrm{\Delta }u-\lambda u+(|x{|}^{-1}\ast |u{|}^2)u-\mathrm{\Delta }(u^2)u-|u{|}^{p-2}u=0,x\in {\mathbb{R}}^3,$$

where $p\in (\frac{10}{3},6)$, $\lambda \in \mathbb{R}$. Firstly, the quasilinear equations are transformed into semilinear equations by making a appropriate change of variables, whose associated variational functionals are well defined in $H_r^1({\mathbb{R}}^3)$. Secondly, by constructing auxiliary functional and combining pohožaev identity, we prove that under constraints, the energy functionals related to the equation have bounded Palais-Smale sequences on each level set. Finally, it is obtained that there are infinitely many normalized radial solutions for this kind of quasilinear Schrödinger-Poisson equations.