$p$-biharmonic Kirchhoff equations with critical Choquard nonlinearity

Abstract

In this article, we deal with the following involving $p$-biharmonic critical Choquard-Kirchhoff equation $$ \left(a+b\left(\int_{\mathbb R^N}|Δu|^p dx\right)^{θ-1}\right) Δ_{p}^{2}u = α\left(|x|^{-μ}*u^{p^*_μ}\right)|u|^{p^*_μ-2}u+ λf(x) |u|^{r-2} u \; \text{in}\; \mathbb R^N, $$ where $a\geq 0$, $b> 0$, $0<μ 2p$, $p\geq 2$, $θ\geq1$, $α$ and $λ$ are positive real parameters, $p_μ^{*}= \frac{p(2N-μ)}{2(N-2p)}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. The function $f \in L^{t}(\mathbb R^N)$ with $t= \frac{p^{*}}{(p^* -r)}$ if $p<r<p^*:=\frac{Np}{N-2p}$ and $t=\infty$ if $r\geq p^{*}$. 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 $λ$ and \(α\) for different values of $r$. The results obtained here are new even for $p-$Laplacian.