| Summary: | In this article, we investigate the multiplicity results of the following biharmonic Choquard system involving critical nonlinearities with sign-changing weight function:
\begin{align*}
\begin{cases}
\Delta^{2}u = \lambda F(x) |u|^{r-2}u+ H(x)\left(\displaystyle\int_{\Omega}\frac{H(y)|v(y)|^{2_\alpha^*}}{|x-y|^{\alpha}}dy\right)|u|^{2_\alpha^*-2}u\;& \text{in}\;\Omega,\\
\Delta^{2}v = \mu G(x) |v|^{r-2}v+ H(x)\left(\displaystyle\int_{\Omega}\frac{H(y)|u(y)|^{2_\alpha^*}}{|x-y|^{\alpha}}dy\right)|v|^{2_\alpha^*-2}v\;& \text{in}\;\Omega,\\
u=v=\nabla u =\nabla v= 0\quad \;& \text{on}\;\partial\Omega,
\end{cases}
\end{align*}
where $\Omega$ is a bounded domain in $\mathbb R^N$ with smooth boundary $\partial \Omega$, $N\geq 5$, $1<r <2$, $0<\alpha<N$, $2_\alpha^*=\frac{2N-\alpha}{N-4}$ is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and $\Delta^2$ denotes the biharmonic operator. The functions $F$, $G$ and $H:\overline{\Omega}\rightarrow \mathbb R$ are sign-changing weight functions satisfying $F$, $G\in L^{\frac{2^*}{2^*-r}}(\Omega)$ and $H\in L^{\infty}(\Omega)$ respectively. By adopting Nehari manifold and fibering map technique, we prove that the system admits at least two nontrivial solutions with respect to parameter $(\lambda, \mu)\in \mathbb R^2_{+} \setminus \{(0, 0)\}$.
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