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Least energy solutions for fractional Kirchhoff type equations involving critical growth

  • * Corresponding author: Yinbin Deng

    * Corresponding author: Yinbin Deng 
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  • We study the following fractional Kirchhoff type equation:

    $ \begin{equation*} \begin{array}{ll} \left \{ \begin{array}{ll} \Big(a+b\int_{ \mathbb{R} ^3}|(-\Delta)^\frac{s}{2}u|^2dx\Big)(-\Delta )^s u+V(x)u = f(u)+|u|^{2^*_s-2}u, \ x\in \mathbb{R} ^3, \\ u\in H^s( \mathbb{R} ^3), \end{array} \right . \end{array} \end{equation*} $

    where $ a, \ b>0 $ are constants, $ 2^*_s = \frac{6}{3-2s} $ with $ s\in(0, 1) $ is the critical Sobolev exponent in $ \mathbb{R} ^3 $, $ V $ is a potential function on $ \mathbb{R} ^3 $. Under some more general assumptions on $ f $ and $ V $, we prove that the given problem admits a least energy solution by using a constrained minimization on Nehari-Pohozaev manifold and monotone method.

    Mathematics Subject Classification: Primary: 35R11; Secondary: 35A15, 35B33.


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