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Ground State Solutions for Fractional p-Kirchhoff Equation with Subcritical and Critical Exponential Growth

  • Ruichang Pei
  • Ying Zhang
  • Jihui Zhang
Article
  • 33 Downloads

Abstract

In this paper, we show the existence of nontrivial ground state solutions of fractional p-Kirchhoff problem
$$\begin{aligned} \left\{ \begin{array}{ll} m\left( \int _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\mathrm{d}x\mathrm{d}y\right) (-\Delta )_p^s u=f(x,u) ~&{}\text {in}~\Omega , \\ u=0 ~&{}\text {in}~{\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. \end{aligned}$$
where \((-\Delta )_p^s\) is the fractional p-Laplacian operator with \(0<s<1<p<\infty \), \(\Omega \) is a bounded domain in \({\mathbb {R}}^N\) with smooth boundary, m is continuous function and the nonlinearity f(xu) has subcritical or critical exponential growth at \(\infty \). For the purpose of obtaining our existence results, we used minimax techniques combined with the fractional Moser–Trudinger inequality.

Keywords

Fractional p-Laplacian Mountain pass theorem Moser–Trudinger inequality Subcritical and critical exponential growth 

Mathematics Subject Classification

34B27 35J60 35B05 

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsTianshui Normal UniversityTianshuiPeople’s Republic of China
  2. 2.School of Mathematical SciencesNanjing Normal UniversityNanjingPeople’s Republic of China

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