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Multiple Solutions of Quasilinear Schrödinger Equations with Critical Growth Via Penalization Method

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Abstract

In this paper, we deal with the quasilinear Schrödinger equation

$$\begin{aligned} -\epsilon ^{2}\Delta u+V(x)u-\epsilon ^2u\Delta (u^2)=h(u)+ u^{22^*-1},\ u>0,\ x\in \mathbb {R}^{N}, \end{aligned}$$

where \(\epsilon >0\) is a small parameter, \(N\ge 3\), V is continuous and h is of subcritical growth. When V satisfies a local condition and h is merely continuous, we obtain the multiplicity and concentration of solutions using the method of Nehari manifold, penalization techniques and Ljusternik–Schnirelmann category theory.

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Acknowledgements

The authors would like to express their sincere gratitude to the referees for careful reading the manuscript and valuable suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Jinling Institute of Technology, Nanjing, 211169, China

    Hui Zhang

  2. School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, 210023, China

    Miao Du

  3. Department of Mathematics, Nanjing Forestry University, Nanjing, 210037, China

    Min Zhu

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  1. Hui Zhang
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  2. Miao Du
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  3. Min Zhu
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Correspondence to Miao Du.

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This work was supported by China Postdoctoral Science Foundation (Nos. 2021M691527, 2020M671531), the National Natural Science Foundation of China (No. 11901284), the Natural Science Foundation of Jiangsu Province (Nos. BK20180814, BK20201382), and the Natural Science Fund for Colleges and Universities in Jiangsu Province (No. 18KJB110009)

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Zhang, H., Du, M. & Zhu, M. Multiple Solutions of Quasilinear Schrödinger Equations with Critical Growth Via Penalization Method. Mediterr. J. Math. 18, 263 (2021). https://doi.org/10.1007/s00009-021-01911-5

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  • DOI: https://doi.org/10.1007/s00009-021-01911-5

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