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Normalized Solutions of Mass Supercritical Kirchhoff Equation with Potential

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Abstract

In this paper, we study the existence of normalized solutions to the following Kirchhoff equation with mass supercritical exponent

$$\begin{aligned} -\left( a+b\int _{{\mathbb {R}}^3}|\nabla u|^2\,\mathrm{{d}}x\right) \Delta u+V(x)u+\lambda u=|u|^{p-2}u, \quad \text {in}\quad {\mathbb {R}}^3, \end{aligned}$$

where \(\lambda \in {\mathbb {R}},p\in \left( \frac{14}{3},6\right) ,V(x)\not \equiv 0.\) Firstly, we prove the existence of normalized solutions under suitable conditions on the potential \(V(x)\ge 0.\) Secondly, when \(V(x)\le 0\) is bounded, we show the existence of mountain pass solutions with positive energy and the nonexistence of solutions with negative energy. Thirdly, when \(V(x)\le 0\) is not too small on a suitable interval, we find a local minimizer of the energy functional with negative energy.

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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. The research of Li Cai is partially supported by the Postgraduate Research and Practice Innovation Program of Jiangsu Province (No. KYCX21_0076). The research of Fubao Zhang are partially supported by National Natural Science Foundation of China (No.11671077) and National Scientific Research Program Cultivation Fund of Chengxian College of Southeast University (No. 2022NCF008).

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Li Cai & Fubao Zhang

  2. Chengxian College, Southeast University, Nanjing, 210088, People’s Republic of China

    Fubao Zhang

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  1. Li Cai
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  2. Fubao Zhang
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Correspondence to Fubao Zhang.

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Cai, L., Zhang, F. Normalized Solutions of Mass Supercritical Kirchhoff Equation with Potential. J Geom Anal 33, 107 (2023). https://doi.org/10.1007/s12220-022-01148-y

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