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Quasilinear Schrödinger Equations with Stein–Weiss Type Nonlinearity and Potential Vanishing at Infinity

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Abstract

In this paper, we consider the following quasilinear Schrödinger equation involving Stein–Weiss type nonlinearity:

$$\begin{aligned}{} & {} -\Delta u+V(x)u-\Delta (u^2)u\\{} & {} \qquad =\frac{1}{|x |^{\alpha }} \left( \int _{\mathbb {R}^N}\frac{G(u(y))}{|y-x |^{\mu }|y |^{\alpha }}dy\right) g(u(x)), \ \ \textrm{in}\ \ \mathbb {R}^N, \end{aligned}$$

where \(N\ge 3\), \(0<\mu <N\), \(\alpha \ge 0\) and \(2\alpha +\mu <\min \{\frac{N+2}{2}, 4\}\) and G is the primitive of function g. The potential \(V: \mathbb {R}^N\rightarrow \mathbb {R}\) may decay to zero at infinity. By using variational methods, penalization technique and \(L^{\infty }\)-estimates, we obtain the existence of a positive solution for the above quasilinear Schrödinger equation.

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Funding

This research was supported by the National Natural Science Foundation of China (No.11901499); Nanhu Scholar Program for Young Scholars of XYNU (No.201912).

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  1. School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, Henan, People’s Republic of China

    Ming-Chao Chen & Yan-Fang Xue

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  1. Ming-Chao Chen
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  2. Yan-Fang Xue
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Correspondence to Yan-Fang Xue.

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Chen, MC., Xue, YF. Quasilinear Schrödinger Equations with Stein–Weiss Type Nonlinearity and Potential Vanishing at Infinity. Qual. Theory Dyn. Syst. 23, 154 (2024). https://doi.org/10.1007/s12346-024-01013-z

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