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Existence of Multiple Solutions for Elliptic Equations with Indefinite Potential

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Abstract

In this paper, the following modified fourth-order Schrödinger equation

$$\begin{aligned} \alpha \Delta ^2 u-\Delta u+V(x)u-u\Delta (u^2)=g(u),\quad&\text {in}~{\mathbb {R}}^{N}\text {,} \end{aligned}$$

and quasilinear Schrödinger equation with \(\alpha =0\) are discussed. The nonlinearity is subquadratic, i.e.,

$$\begin{aligned} \lim _{|t|\rightarrow \infty }\frac{g(t)}{t^2}=0\text {,} \end{aligned}$$

and the potential V is indefinite in sign. By variational methods, we will prove the existence of multiple solutions if \(\alpha \ne 0\) and \(N\le 6\) or \(\alpha =0\) and \(N\ge 3\).

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Acknowledgements

The authors would like to thank the reviewers for careful reading of the manuscript and for valuable comments and suggestions.

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  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Lifeng Yin & Shuai Jiang

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  1. Lifeng Yin
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Correspondence to Lifeng Yin.

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Yin, L., Jiang, S. Existence of Multiple Solutions for Elliptic Equations with Indefinite Potential. Qual. Theory Dyn. Syst. 23, 33 (2024). https://doi.org/10.1007/s12346-023-00888-8

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