$p$-biharmonic Kirchhoff equations with critical Choquard nonlinearity

Abstract

In this article, we deal with the following involving $p$-biharmonic critical Choquard-Kirchhoff equation $$\left(a+b{\left({\int }_{{\mathbb{R}}^N}|\Delta u{|}^{p}dx\right)}^{\theta -1}\right){\Delta }_p^{2}u=\alpha \left(|x{|}^{-\mu }\ast u^{p_{\mu }^{\ast }}\right)|u{|}^{p_{\mu }^{\ast }-2}u+\lambda f(x)|u{|}^{r-2}u\text{in}{\mathbb{R}}^N,$$ where $a\ge 0$, $b>0$, $0<\mu <N$, $N>2p$, $p\ge 2$, $\theta \ge 1$, $\alpha$ and $\lambda$ are positive real parameters, $p_{\mu }^{\ast }=\frac{p(2N-\mu )}{2(N-2p)}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. The function $f\in L^t({\mathbb{R}}^N)$ with $t=\frac{p^{\ast }}{(p^{\ast }-r)}$ if $p<r<p^{\ast }:=\frac{Np}{N-2p}$ and $t=\infty$ if $r\ge p^{\ast }$. We first prove the concentration compactness principle for the $p$-biharmonic Choquard-type equation. Then using the variational method together with the concentration-compactness, we established the existence and multiplicity of solutions to the above problem with respect to parameters $\lambda$ and \(α\) for different values of $r$. The results obtained here are new even for $p-$Laplacian.