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

Abstract

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: \begin{equation*} \begin{array}{lll} M(\|u\|_μ^{p})(μ(-Δ)^{s}_{p}u+V(ξ)|u|^{p-2}u)= f(ξ,u)+\int_{\mathbb{H}^N}\frac{|u(η)|^{Q_λ^{\ast}}}{|η^{-1}ξ|^λ}dη|u|^{Q_λ^{\ast}-2}u &\mbox{in}\ \mathbb{H}^N, \\ \end{array} \end{equation*} where $(-\Delta {)}_p^s$ 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.

References in corpus (1)

• Brezis-Nirenberg type result for Kohn Laplacian with critical Choquard Nonlinearity