Biharmonic system with Hartree-type critical nonlinearity

Abstract

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 $(\lambda, \mu)\in \mathbb R^2_{+} \setminus \{(0, 0)\}$.

Keywords

• biharmonic system
• sign-changing weight function
• nehari manifold
• hatree-type critical nonlinearity