On multiplicity and concentration of solutions for fractional $p$-Laplace Choquard-Kirchhoff equations

Abstract

In this paper, we consider theChoquard-Kirchhoff involving fractional $p$-Laplace equation of the form: \begin{align*}M \Big (\varepsilon^{sp-N}[u]^p_{s,p}+\varepsilon^{-N}\int_{\mathbb{R}^{N}}V(x)|u|^pdx \Big )&\big(\varepsilon^{sp}(-\Delta)_p^su+V(x)|u|^{p-2}u)\big)\\& = \varepsilon^{\mu -N}(\frac {1}{|x|^\mu}*F(u))f(u) \end{align*} in $\mathbb{R}^N,$ where $\varepsilon > 0$ is a parameter, $s\in (0 , 1)$,$0 < \mu < ps$, $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $M$ represents Kirchhoff function, $V$ is a continuous potential function and $f$ is a continuous function involvingsubcritical growth.With the help of well-known penalization methods and Ljusternik-Schnirelmann theory, we obtain the existence, multiplicity and concentration of solutions forthe above equations.