$p$-biharmonic Kirchhoff equations with critical Choquard nonlinearity

Abstract

In this article, we deal with the following involving $p$-biharmonic critical Choquard-Kirchhoff equation $$ \left(a+b\left(\int_{\mathbb R^N}|\Delta u|^p dx\right)^{\theta-1}\right) \Delta_{p}^{2}u = \alpha \left(|x|^{-\mu}*u^{p^*_\mu}\right)|u|^{p^*_\mu-2}u+ \lambda f(x) |u|^{r-2} u \; \text{in}\; \mathbb R^N, $$ where $a\geq 0$, $b> 0$, $0<\mu<N$, $N>2p$, $p\geq 2$, $\theta\geq1$, $\alpha$ and $\lambda$ are positive real parameters, $p_{\mu}^{*}= \frac{p(2N-\mu)}{2(N-2p)}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. The function $f \in L^{t}(\mathbb R^N)$ with $t= \frac{p^{*}}{(p^* -r)}$ 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 $\lambda$ and \(\alpha\) for different values of $r$. The results obtained here are new even for $p-$Laplacian.