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Existence and asymptotic behavior of positive solutions for a class of locally superlinear Schrödinger equation

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This paper treats the existence of positive solutions of \(-\Delta u + V(x) u = \uplambda f(u)\) in \({\mathbb {R}}^N\). Here \(N \ge 1\), \(\uplambda > 0\) is a parameter and f(u) satisfies conditions only in a neighborhood of \(u=0\). We shall show the existence of positive solutions with potential of trapping type or \({\mathcal {G}}\)-symmetric potential where \({\mathcal {G}} \subset O(N)\). Our results extend previous results (Adachi and Watanabe in J Math Anal Appl 507:125765, 2022; Costa and Wang in Proc Am Math Soc 133(3):787–794, 2005; do Ó et al. in J Math Anal Appl 342:432–445, 2008) as well as we also study the asymptotic behavior of a family \((u_\uplambda )_{\uplambda \ge \uplambda _0}\) of positive solutions as \(\uplambda \rightarrow \infty \).

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Numbers JP19K03590, JP19H01797, JP18K03362, JP21K03317 and by JSPS-NSFC joint research project “Variational study of nonlinear PDEs" and by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.

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Authors and Affiliations

  1. Department of Mathematical and Systems Engineering, Faculty of Engineering, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu, 432-8561, Japan

    Shinji Adachi

  2. Department of Mathematics, Faculty of Science and Technology, Keio University, Yagami Campus: 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa, 223-8522, Japan

    Norihisa Ikoma

  3. Department of Mathematics, Faculty of Science, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto-City, 603-8555, Japan

    Tatsuya Watanabe

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  1. Shinji Adachi
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  2. Norihisa Ikoma
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Correspondence to Shinji Adachi.

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Adachi, S., Ikoma, N. & Watanabe, T. Existence and asymptotic behavior of positive solutions for a class of locally superlinear Schrödinger equation. manuscripta math. 172, 933–970 (2023). https://doi.org/10.1007/s00229-022-01428-5

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