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Positive Ground State Solutions for Generalized Quasilinear Schrödinger Equations with Critical Growth

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This paper concerns the existence of positive ground state solutions for generalized quasilinear Schrödinger equations in \({\mathbb {R}}^{N}\) with critical growth which arise from plasma physics, as well as high-power ultrashort laser in matter. By applying a variable replacement, the quasilinear problem reduces to a semilinear problem which the associated functional is well defined in the Sobolev space \(H^1({\mathbb {R}}^N)\). We use the method of Nehari manifold for the modified equation, establish the minimax characterization, and then prove that each Palais-Smale sequence of the associated energy functional is bounded. By combining Lions’s concentration-compactness lemma together with some classical arguments developed by Brézis and Nirenberg (Commun Pure Appl Math 36:437–477, 1983), we obtain that the bounded Palais-Smale sequence has a nonvanishing behavior. Finally, we establish the existence of a positive ground state solution under some appropriate assumptions.

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Acknowledgements

The authors sincerely thank the referees for very careful reading and many valuable comments, which led to much improvement in earlier version of this paper. This work is partially supported by NSFC Grants (12225103, 12071065, and 11871140) and the National Key Research and Development Program of China (2020YFA0713602 and 2020YFC1808301).

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Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

    Xin Meng & Shuguan Ji

  2. School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, People’s Republic of China

    Shuguan Ji

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  1. Xin Meng
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Correspondence to Shuguan Ji.

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Meng, X., Ji, S. Positive Ground State Solutions for Generalized Quasilinear Schrödinger Equations with Critical Growth. J Geom Anal 33, 372 (2023). https://doi.org/10.1007/s12220-023-01429-0

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