Kirchhoff equations with Choquard exponential type nonlinearity involving the fractional Laplacian

Abstract

In this article, we deal with the existence of non-negative solutions of the class of following non local problem $$ \left\{ \begin{array}{lr} \quad - M\left(\displaystyle\int_{\mathbb R^n}\int_{\mathbb R^{n}} \frac{|u(x)-u(y)|^{\frac{n}{s}}}{|x-y|^{2n}}~dxdy\right) (-\Delta)^{s}_{n/s} u=\left(\displaystyle\int_{\Omega}\frac{G(y,u)}{|x-y|^{\mu}}~dy\right)g(x,u) \; \text{in}\; \Omega,\\ \quad \quad u =0\quad\text{in} \quad \mathbb R^n \setminus \Omega, \end{array} \right. $$ where $(-\Delta)^{s}_{n/s}$ is the $n/s$-fractional Laplace operator, $n\geq 1$, $s\in(0,1)$ such that $n/s\geq 2$, $\Omega\subset \mathbb R^n$ is a bounded domain with Lipschitz boundary, $M:\mathbb R^+\rightarrow \mathbb R^+$ and $g:\Omega\times\mathbb R\rightarrow \mathbb R$ are continuous functions, where $g$ behaves like $\exp({|u|^{\frac{n}{n-s}}})$ as $|u|\rightarrow\infty$.