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# International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

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Volume 19, Issue 2

# Multiplicity Results for Degenerate Fractional $\mathbit{p}$-Laplacian Problems with Critical Growth

Li Wang
/ Jixiu Wang
• Corresponding author
• School of Mathematics and Statistics, Hubei University of Arts and Science, Xiangyang 441053, China
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Published Online: 2018-03-07 | DOI: https://doi.org/10.1515/ijnsns-2017-0195

## Abstract

In this paper, we deal with the existence of multiple nontrivial solutions for the following fractional $p$-Laplacian Kirchhoff problems $\left\{\begin{array}{ll}& \left({\iint }_{{\mathbb{R}}^{2N}}\frac{|u\left(x\right)-u\left(y\right){|}^{p}}{|x-y{|}^{N+ps}}\mathrm{d}x\mathrm{d}y{\right)}^{\theta -1}\left(-\mathrm{\Delta }{\right)}_{p}^{s}u=\lambda |u{|}^{\theta p-2}u\\ & \phantom{\rule{1em}{0ex}}+|u{|}^{{p}_{s}^{\ast }-2}u\text{in}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\Omega },\\ & u=0\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\text{in}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathbb{R}}^{N}\setminus \mathrm{\Omega },\end{array}\right)$

where parameter $\lambda >0$ belongs to some left neighbourhood of the eigenvalue of the nonlocal operator

$\left({\iint }_{{\mathbb{R}}^{2N}}\frac{|u\left(x\right)-u\left(y\right){|}^{p}}{|x-y{|}^{N+ps}}\mathrm{d}x\mathrm{d}y{\right)}^{\theta -1}\left(-\mathrm{\Delta }{\right)}_{p}^{s}.$

The main feature and difficulty of our problems is the fact that the problem is degenerate.

MSC 2010: 35B33; 35R11; 45C05; 58E05

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Accepted: 2018-02-21

Published Online: 2018-03-07

Published in Print: 2018-04-25

The first author was supported by National Natural Science Foundation of China (11561024, 11701178), the second author was was supported by National Natural Science Foundation of China (11501186).

Competing interests The authors declare that they have no competing interests.

Authors contributions All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 2, Pages 215–222, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339,

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