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Existence of positive solutions to Kirchhoff equations with vanishing potentials and general nonlinearity

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Abstract

We study the existence of positive solutions to the following Kirchhoff type equation with vanishing potential and general nonlinearity:

$$\begin{aligned} \left\{ \begin{aligned}&-(\varepsilon ^2a+\varepsilon b{\int _{\mathbb {R}^3}}{|\nabla v|}^{2})\Delta v+V(x)v=f(v), ~~~~x\in \mathbb {R}^3, \\&v>0,~~~v\in H^{1}(\mathbb {R}^3), \end{aligned} \right. \end{aligned}$$

where \(\varepsilon >0\) is a small parameter, \(a,b>0\) are constants and the potential V can vanish, i.e., the zero set of V, \(\mathcal {Z}:=\{x\in \mathbb {R}^3|V(x)=0\}\) is non-empty. In our case, the method of Nehari manifold does not work any more. We first make a truncation of the nonlinearity and prove the existence of solutions for the equation with truncated nonlinearity, then by elliptic estimates, we prove that the solution of truncated equation is just the solution of our original problem for sufficiently small \(\varepsilon >0\).

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Acknowledgements

The authors would like to express sincere thanks to the anonymous referees for their carefully reading the manuscript and valuable comments and suggestions.

Funding

Supported by the National Natural Science Foundation of China (11771428, 11926335), Natural Science Foundation of Shandong Province (ZR2018MA009), Project of Shandong Province Higher Educational Science and Technology Program (J18KB103, J18KB109).

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Authors and Affiliations

  1. School of Mathematics, Qilu Normal University, Jinan, 250013, People’s Republic of China

    Dongdong Sun

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Zhitao Zhang

  3. School of Mathematical Sciences, University of Chinese, Academy of Sciences, Beijing, 100049, People’s Republic of China

    Zhitao Zhang

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  1. Dongdong Sun
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  2. Zhitao Zhang
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Correspondence to Dongdong Sun.

Additional information

This article is part of the topical collection dedicated to Prof. Dajun Guo for his 85th birthday, edited by Yihong Du, Zhaoli Liu, Xingbin Pan, and Zhitao Zhang.

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Sun, D., Zhang, Z. Existence of positive solutions to Kirchhoff equations with vanishing potentials and general nonlinearity. SN Partial Differ. Equ. Appl. 1, 8 (2020). https://doi.org/10.1007/s42985-020-00010-6

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  • DOI: https://doi.org/10.1007/s42985-020-00010-6

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