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Existence of Solutions to a Class of p-Kirchhoff Equations via Morse Theory

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Abstract

This paper is devoted to the following p-Kirchhoff type of problems:

$$\begin{aligned} \left\{ \begin{array}{ll} -(a+b\int _{\Omega }|\nabla u|^{p}\,\text{ d }x)\Delta _{p} u=-\lambda |u|^{q-2}u+f(x,u),x\in \Omega \\ u=0,x\in \partial \Omega . \end{array} \right. \end{aligned}$$

Without assuming the standard subcritical polynomial growth condition ensuring the compactness of a bounded (P.S.) sequence, we show that the Dirichlet boundary value problem has at least a weak nontrivial solution by using Morse theory.

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Acknowledgements

The authors thank the referee for valuable suggestions.

Funding

This paper is supported by National Natural Science Foundation of China (11671331), Natural Science Foundation of Fujian Province (no. 2020J01708) and National Foundation Training Program of Jimei University (ZP2020057).

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  1. School of Science, Jimei University, Xiamen, 361021, People’s Republic of China

    BiYun Tang & YongYi Lan

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  1. BiYun Tang
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The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

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Correspondence to YongYi Lan.

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Tang, B., Lan, Y. Existence of Solutions to a Class of p-Kirchhoff Equations via Morse Theory. Mediterr. J. Math. 18, 114 (2021). https://doi.org/10.1007/s00009-021-01749-x

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