Existence and multiplicity of solutions for critical Kirchhoff-Choquard equations involving the fractional p-Laplacian on the Heisenberg group

In this paper, we study existence and multiplicity of solutions for the following Kirchhoff-Choquard type equation involving the fractional $p$-Laplacian on the Heisenberg group: $M(\|u\|_\mu^{p})(\mu(-\Delta)^{s}_{p}u+V(\xi)|u|^{p-2}u)= f(\xi,u)+\int_{\mathbb{H}^N}\frac{|u(\eta)|^{Q_\lambda^{\ast}}}{|\eta^{-1}\xi|^\lambda}d\eta|u|^{Q_\lambda^{\ast}-2}u$ in $\mathbb{H}^N$, where $(-\Delta)^{s}_{p}$ is the fractional $p$-Laplacian on the Heisenberg group $\mathbb{H}^N$, $M$ is the Kirchhoff function, $V(\xi)$ is the potential function, $0 < s < 1$, $1 < p < \frac{N}{s}$, $\mu > 0$, $f(\xi,u)$ is the nonlinear function, $0 < \lambda < Q$, $Q=2N+2$, and $Q_\lambda^{\ast}=\frac{2Q-\lambda}{Q-2}$ is the Sobolev critical exponent. Using the Krasnoselskii genus theorem, the existence of infinitely many solutions is obtained if $\mu$ is sufficiently large. In addition, using the fractional version of the concentrated compactness principle, we prove that problem has $m$ pairs of solutions if $\mu$ is sufficiently small. As far as we know, the results of our study are new even in the Euclidean case.

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