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Ground State Solutions for Generalized Quasilinear Schrödinger Equations with Critical Growth

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Abstract

In this paper, we study the following generalized quasilinear Schrödinger equation with critical growth

$$\begin{aligned} -\hbox {div}(g^2(u)\nabla u)+g(u)g^\prime (u)|\nabla u|^2+V(x)u =\lambda f(x,u)+|u|^{\alpha 2^*-2}u, \quad x \in {\mathbb {R}}^N, \end{aligned}$$

where \(\lambda >0\), \(N\ge 3\), \(g: {\mathbb {R}}\rightarrow {\mathbb {R}}^+\) is a \(C^1\) even function, \(g(0)=1\), \(g^\prime (s)\ge 0\) for all \(s\ge 0\), \(g(s)=\beta |s|^{\alpha -1}+O(|s|^{\gamma -1})\) as \(s\rightarrow \infty \) for some constants \(\alpha \in [1,2], \ \beta >0, \ \gamma <\alpha \) and \((\alpha -1)g(s)\ge g^\prime (s)s\) for all \(s\ge 0\). Under some suitable conditions on the potential and nonlinearity, we obtain the existence of ground state solutions for large \(\lambda \) by using dual approach and Nehari manifold method.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (12261031, 12271152, 12161033), the Yunnan Province Applied Basic Research for General Project (2019FB001), Youth Outstanding-notch Talent Support Program in Yunnan Province and the Project Funds of Xingdian Talent Support Program, the Natural Science Foundation of Hunan Province (2021JJ30189, 2022JJ30200), the Key project of Scientific Research Project of Department of Education of Hunan Province (21A0387), the China Scholarship Council (201908430218) for visiting the University of Craiova (Romania). Jian Zhang would like to thank the China Scholarship Council and the Embassy of the People’s Republic of China in Romania.

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Authors and Affiliations

  1. Department of Mathematics, Honghe University, Mengzi, 661100, Yunnan, People’s Republic of China

    Quanqing Li

  2. College of Science, Hunan University of Technology and Business, Changsha, 410205, Hunan, People’s Republic of China

    Jian Zhang

  3. Key Laboratory of Hunan Province for Statistical Learning and Intelligent Computation, Changsha, 410205, Hunan, People’s Republic of China

    Jian Zhang

  4. Department of Mathematics, University of Craiova, 200585, Craiova, Romania

    Jian Zhang

  5. China-Romania Research Center in Applied Mathematics, 200585, Craiova, Romania

    Jian Zhang

  6. School of Mathematics and Physics, North China Electric Power University, Beijing, Beijing, 102206, People’s Republic of China

    Jianjun Nie

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  1. Quanqing Li
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Correspondence to Jian Zhang.

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Li, Q., Zhang, J. & Nie, J. Ground State Solutions for Generalized Quasilinear Schrödinger Equations with Critical Growth. Qual. Theory Dyn. Syst. 21, 137 (2022). https://doi.org/10.1007/s12346-022-00667-x

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