关键词: Hardy-Littlewood-Sobolev inequality, concentration-compactness principle, variational method, Fractional $p$-Laplacian operators, multiple solutions

Abstract: In this paper, we consider the following new Kirchhoff-type equations involving the fractional $p$-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity:\begin{eqnarray*} && \left (a+ b\iint _{\mathbb{R}^{2N}} \frac{|u (x)-u (y)|^p}{|x-y|^{N + ps}} {\rm d}x {\rm d}y\right)^{p-1} (-\Delta)_p^s u + \lambda V(x)|u|^{p-2}u\\ &=& \bigg(\int_{\mathbb{R}^N} \frac{|u|^{p^*_{\mu,s}}}{|x-y|^\mu}{\rm d}y\bigg)|u|^{p^*_{\mu,s}-2}u, \; x \in \mathbb{R}^N, \end{eqnarray*} where $(-\Delta)_p^s$ is the fractional $p$-Laplacian with $0 < s < 1 < p$, $0 < \mu < N$, $N > ps$, $a,b>0$, $\lambda>0$ is a parameter, $V:\mathbb{R}^N \to \mathbb{R}^+$ is a potential function, $\theta \in[1, 2^*_{\mu,s})$ and $p^*_{\mu,s}=\frac{pN-p\frac{\mu}{2}}{N-ps}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We get the existence of infinitely many solutions for the above problem by using the concentration compactness principle and Krasnoselskii's genus theory. To the best of our knowledge, our result is new even in Choquard-Kirchhoff-type equations involving the $p$-Laplacian case.

Key words: Hardy-Littlewood-Sobolev inequality, concentration-compactness principle, variational method, Fractional $p$-Laplacian operators, multiple solutions