Fractional Choquard-Kirchhoff problems with critical nonlinearity and Hardy potential
2021
In this paper, we investigate the following fractional p-Kirchhoff type problem $$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b[u]_{s,p}^{p(\theta -1)}\right) (-\Delta )^s_pu = \Big ({\mathcal {I}}_\mu *|u|^q\Big )|u|^{q-2}u+\frac{|u|^{p_{\alpha }^*-2}u}{|x|^\alpha },\ u>0, &{}\text{ in }\ \Omega ,\\ u=0, \ &{} \mathrm{in}\ {\mathbb {R}}^N\backslash \Omega , \end{array} \right. \end{aligned}$$
where $$[u]_{s,p}^{p}=\displaystyle \iint _{{\mathbb {R}}^{2N}} \frac{|u(x) - u(y)|^{p}}{|x - y|^{N+ps}}\, dxdy$$
, $$\Omega $$
is a bounded smooth domain of $${\mathbb {R}}^N$$
containing 0 with Lipschitz boundary, $$(-\Delta )_{p}^{s}$$
denotes the fractional p-Laplacian, $$0\le \alpha1$$
, $$a\ge 0$$
, $$b>0$$
, $$1<\theta \le p_\alpha ^*/ p$$
, $$p_\alpha ^*=\frac{(N-\alpha )p}{N-ps}$$
is the fractional critical Hardy-Sobolev exponent, $${\mathcal {I}}_\mu (x)=|x|^{-\mu }$$
is the Riesz potential of order $$\mu \in (0,\min \{N,2ps\})$$
, $$q\in (1, Np/(N-ps))$$
satisfies some restrictions. By the concentration-compactness principle and mountain pass theorem, we obtain the existence of a positive weak solution for the above problem as q satisfies suitable ranges.
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