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Multiple positive solutions to critical p-Laplacian equations with vanishing potential

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Abstract

This paper deals with the following p-Laplacian equation

$$\begin{aligned} -\varepsilon ^{p}\Delta _{p}u+V(x)|u|^{p-2}u=|u|^{p^{*}-2}u,\quad u\in D^{1,p}({\mathbb {R}}^N), \end{aligned}$$

where \(p\in (1,N)\), p-Laplacian operator \(\Delta _{p}{:}{=}\)div\((|\nabla u|^{p-2}\nabla u) \), \(p^{*}=Np/(N-p)\), \(\varepsilon \) is a positive parameter, \(V(x)\in L^{{N}/{p}}({\mathbb {R}}^N)\cap L^{\infty }_{loc}({\mathbb {R}}^N)\) and V(x) is assumed to be zero in some region of \({\mathbb {R}}^N\), which means it is of the vanishing potential case. In virtue of Ljusternik–Schnirelman theory of critical points, we succeed in proving the multiplicity of positive solutions. This result generalizes the result for semilinear Schrödinger equation by Chabrowski and Yang (Port. Math. 57 (2000), 273–284) to p-Laplacian equation.

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Acknowledgements

The authors would like to express their gratitude to the reviewers for careful reading and helpful suggestions which led to an improvement of the original manuscript. The authors also thank Professor Shuangjie Peng very much for helpful suggestions on the present paper. The work is supported by the National Natural Science Foundation of China (No. 11901222, 12071169), the China Postdoctoral Science Foundation (No. 2021M690039) and the excellent doctorial dissertation cultivation grant (No. 2019YBZZ057) from Central China Normal University.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Lun Guo

  2. College of Science, Wuhan University of Science and Technology, Wuhan, 430065, People’s Republic of China

    Qi Li

  3. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, People’s Republic of China

    Qi Li

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  1. Lun Guo
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  2. Qi Li
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Correspondence to Qi Li.

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Guo, L., Li, Q. Multiple positive solutions to critical p-Laplacian equations with vanishing potential. Z. Angew. Math. Phys. 72, 167 (2021). https://doi.org/10.1007/s00033-021-01598-4

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