## Kirchhoff Biharmonic System with Choquard Nonlinearity and Singular Weights

Published: 2024-04-08

Page: 8-28

Luying Wu *

School of Mathematics and Statistics, Southwest University, 400715, China.

*Author to whom correspondence should be addressed.

### 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.

Keywords: Biharmonic equation, Kirchhoff type system, choquard nonlinearity, critical exponential growth

#### How to Cite

Wu, L. (2024). Kirchhoff Biharmonic System with Choquard Nonlinearity and Singular Weights. Asian Journal of Mathematics and Computer Research, 31(2), 8–28. https://doi.org/10.56557/ajomcor/2024/v31i28635

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