Kirchhoff Biharmonic System with Choquard Nonlinearity and Singular Weights

Abstract

The aim of this paper is to find the existence of solutions for the following Kirchhoff type biharmonic system with exponential nonlinearity and singular weights \(\begin{cases}m\left(\|u\|^2+\|v\|^2\right) \Delta^2 u=\left[I_\mu * \frac{F(x, u, v)}{|x|^\alpha}\right] \frac{f_1(x, u, v)}{|x|^\alpha} & \text { in } \Omega \\ m\left(\|u\|^2+\|v\|^2\right) \Delta^2 v=\left[I_\mu * \frac{F(x, u, v)}{|x|^\alpha}\right] \frac{f_2(x, u, v)}{|x|^\alpha} & \text { in } \Omega \\ u=0, \quad v=0, \quad \nabla u=\mathbf{0}, \quad \nabla v=\mathbf{0} & \text { on } \partial \Omega\end{cases}S\) where \(\Omega\) is a bounded domain in \(\mathbb{R}^4\) containing the origin with smooth boundary, \(\mu \in(0,4), 0<\alpha<\frac{\mu}{2}\), \(I_\mu(x)=\frac{1}{|x|^4-\mu}, m\) is a Kirchhoff type function, \(\|u\|^2=\int_{\Omega}|\Delta u|^2 d x, f_i\) behaves like \(e^{\beta_{0 s^2}}\) when \(|s| \rightarrow \infty\) for some \(\beta_0>0\), and there is \(C^1\) function \(F: \mathbb{R}^2 \rightarrow \mathbb{R}\) such that \(\left(\frac{\partial F(x, u, v)}{\partial u}, \frac{\partial F(x, u, v)}{\partial v}\right)=\left(f_1(x, u, v), f_2(x, u, v)\right)\). We establish sufficient conditions for the solutions of the above system by using variational methods with Adams inequality.