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On a class of Kirchhoff type problems

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Abstract

In this paper we consider the following Kirchhoff type problem:

$$(\mathcal{K}) \quad \left(1 + \lambda \int\limits_{\mathbb{R}^3}\big(|\nabla u|^2 + V(y)u^2dy\big)\right)[-\Delta u + V(x)u] = |u|^{p-2}u, \quad {\rm in} \, \mathbb{R}^3,$$

where \({p\in (2, 6)}\), λ > 0 is a parameter, and V(x) is a given potential. Some existence and nonexistence results are obtained by using variational methods. Also, the “energy doubling” property of nodal solutions of \({(\mathcal{K})}\) is discussed in this paper.

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Authors and Affiliations

  1. Department of Mathematics, Soochow University, Suzhou, 215006, Jiangsu, People’s Republic of China

    Yisheng Huang & Zeng Liu

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  1. Yisheng Huang
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  2. Zeng Liu
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Correspondence to Zeng Liu.

Additional information

Supported by Natural Science Foundation of China (11071180, 11171247) and CPRI Project of Jiangsu Province (CXZZ12_0802).

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Huang, Y., Liu, Z. On a class of Kirchhoff type problems. Arch. Math. 102, 127–139 (2014). https://doi.org/10.1007/s00013-014-0618-4

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  • DOI: https://doi.org/10.1007/s00013-014-0618-4

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