Existence of infinitely many normalized radial solutions for a class of quasilinear Schrödinger-Poisson equations in R³

Abstract

In this paper, we study the existence of infinitely many normalized radial solutions for the following quasilinear Schrödinger-Poisson equations: $ \begin{equation*} -\Delta u-\lambda u+(|x|^{-1}*|u|^2)u-\Delta(u^2)u-|u|^{p-2}u = 0,\; x\in\mathbb{R}^3, \end{equation*} $ 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.

Keywords

• mountain pass geometry
• pohožaev identity
• normalized radial solutions