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Existence of Solutions for a Quasilinear Schrödinger Equation with Potential Vanishing

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Abstract

We study the following quasilinear Schrödinger equation

$$ - \Delta u + V(x)u - \Delta ({u^2})u = K(x)g(u),\,\,\,\,\,\,\,\,x \in {\mathbb{R}^3},$$

where the nonlinearity g(u) is asymptotically cubic at infinity, the potential V(x) may vanish at infinity. Under appropriate assumptions on K(x), we establish the existence of a nontrivial solution by using the mountain pass theorem.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xinyang Normal University, Henan, 464000, China

    Yan-fang Xue, Jian-xin Han & Xin-cai Zhu

Authors
  1. Yan-fang Xue
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  2. Jian-xin Han
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  3. Xin-cai Zhu
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Correspondence to Yan-fang Xue.

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Conflict of Interest

The authors declare no conflict of interest.

The project is supported by the National Natural Science Foundation of China (No.11901499 and No.11901500) and Nanhu Scholar Program for Young Scholars of XYNU (No.201912).

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Xue, Yf., Han, Jx. & Zhu, Xc. Existence of Solutions for a Quasilinear Schrödinger Equation with Potential Vanishing. Acta Math. Appl. Sin. Engl. Ser. 39, 696–706 (2023). https://doi.org/10.1007/s10255-023-1083-2

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  • DOI: https://doi.org/10.1007/s10255-023-1083-2

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