For EDITORS

For READERS

All Issues

Vol.10, 2020
Vol.9, 2019
Vol.8, 2018
Vol.7, 2017
Vol.6, 2016
Vol.5, 2015
Vol.4, 2014
Vol.3, 2013
Vol.2, 2012
Vol.1, 2011
Positive and Sign-Changing Solutions for the Fractional Kirchhoff Equation with Critical Growth
Qiu-Ying Peng,Zeng-Qi Ou,Ying Lv
Keywords:Fractional Kirchhoff equation, Variational method, Positive solution, Sign-changing solution
Abstract:
      We are interested in the existence of positive and sign-changing solutions for the fractional Kirchhoff equation $$ \left(a+b\int_{\mathbb{R}^3}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s}u +V(x)u=h(x)|u|^{p-2}u+|u|^{2^*_{s}-2}u\ \ \ \text{in}\ \mathbb{R}^3, $$ where $a, b>0,\ s\in(\frac{3}{4}, 1)$ and $p\in(4,2^*_{s})$ with $2^*_{s}=\frac{6}{3-2s}.$ Under some mild conditions on $V(x)$ and $h(x)$, using variational methods, we prove the existence of positive ground state solutions and least energy sign-changing solutions.