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Positive Solutions for a Class of Quasilinear Schrödinger Equations with Two Parameters

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Abstract

In this paper, we consider the following quasilinear Schrödinger equation

$$\begin{aligned} -\Delta u+V(x)u+\frac{\kappa }{2}\Delta (u^2)u=\lambda f(u)+h(u),\,\, x\in {\mathbb {R}}^N, \end{aligned}$$

where \(\lambda ,\kappa >0\), \(N\ge 3\) and \(2^*=\frac{2N}{N-2}\). By using a change of variable, we obtain the existence of positive solutions for this problem with subcritical nonlinearities by using the mountain pass theorem and Moser iterative method. Our results extend and supplement some other related literatures.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11461043, 11661053, 11771198) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (20161BAB201009, 20181BAB201003), and the Outstanding Youth Scientist Foundation Plan of Jiangxi (2017BCB23004).

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Authors and Affiliations

  1. Department of Mathematics, Nanchang University, Nanchang, 330031, Jiangxi, People’s Republic of China

    Jianhua Chen, Xianjiu Huang & Chuanxi Zhu

  2. School of Traffic and Transportation Engineering, Central South University, Changsha, 410075, Hunan, People’s Republic of China

    Qingfang Wu

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  1. Jianhua Chen
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  2. Qingfang Wu
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  3. Xianjiu Huang
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  4. Chuanxi Zhu
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Correspondence to Xianjiu Huang.

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Communicated by Yong Zhou.

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Chen, J., Wu, Q., Huang, X. et al. Positive Solutions for a Class of Quasilinear Schrödinger Equations with Two Parameters. Bull. Malays. Math. Sci. Soc. 43, 2321–2341 (2020). https://doi.org/10.1007/s40840-019-00803-y

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  • DOI: https://doi.org/10.1007/s40840-019-00803-y

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