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Least Energy Solutions of the Schrödinger–Kirchhoff Equation with Linearly Bounded Nonlinearities

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Abstract

In this paper, we consider the following Schrödinger–Kirchhoff equation

$$ -\left( a+b \displaystyle \int _{\mathbb {R}^N} |\nabla u|^2 \ d x \right) \Delta u+V(x)u=f(x,u), \ \hbox {in}\ \mathbb {R}^N, $$

where \(N\ge 3\), a and b are positive parameters, V(x) is a positive and continuous potential. Under some suitable assumptions on the nonlinearity f(x, u) which allow it is linearly bounded at infinity, the existence of least energy solutions and their asymptotic behavior as \(b\rightarrow 0\) are established via variational methods. The nonexistence of nontrivial solutions is also established for large b.

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Acknowledgements

We are grateful to the anonymous referees for valuable comments and suggestions, which helped us to improve our manuscript. Y. Liu was supported by National Natural Science Foundation of China (No. 12101020). L. Zhao was supported by National Natural Science Foundation of China (No. 12171014, 12171326).

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  1. School of Mathematics and Statistics, Beijing Technology and Business University, Beijing, 100048, People’s Republic of China

    Yanyan Liu & Leiga Zhao

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  1. Yanyan Liu
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YL: wrote the main manuscript text. LZ: proposed the methodology and performed the analysis. All authors reviewed the manuscript.

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Correspondence to Leiga Zhao.

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Liu, Y., Zhao, L. Least Energy Solutions of the Schrödinger–Kirchhoff Equation with Linearly Bounded Nonlinearities. Qual. Theory Dyn. Syst. 23, 4 (2024). https://doi.org/10.1007/s12346-023-00859-z

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