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

Jinfu Yang; Wenmin Li; Wei Guo; Jiafeng Zhang; Jinfu Yang; Wenmin Li; Wei Guo; Jiafeng Zhang

doi:10.3934/math.20221059

AIMS Mathematics

2022, Volume 7, Issue 10: 19292-19305. doi: 10.3934/math.20221059

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

School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China

Received: 11 July 2022 Revised: 14 August 2022 Accepted: 23 August 2022 Published: 31 August 2022
MSC : 35A15, 35B38, 49J35

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.
- mountain pass geometry,
- pohožaev identity,
- normalized radial solutions
Citation: Jinfu Yang, Wenmin Li, Wei Guo, Jiafeng Zhang. Existence of infinitely many normalized radial solutions for a class of quasilinear Schrödinger-Poisson equations in $ \mathbb{R}^3 $[J]. AIMS Mathematics, 2022, 7(10): 19292-19305. doi: 10.3934/math.20221059

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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.

References

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