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 $$\{\begin{array}{l}-M\left({\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\frac{|u(x)-u(y){|}^{\frac{n}{s}}}{|x-y{|}^{2n}}dxdy\right)(-\Delta {)}_{n/s}^{s}u=\left({\int }_{\Omega }\frac{G(y,u)}{|x-y{|}^{\mu }}dy\right)g(x,u)\text{in}\Omega ,\\ u=0\text{in}{\mathbb{R}}^n\setminus \Omega ,\end{array}$$ where $(-\Delta {)}_{n/s}^s$ is the $n/s$-fractional Laplace operator, $n\ge 1$, $s\in (0,1)$ such that $n/s\ge 2$, $\Omega \subset {\mathbb{R}}^n$ is a bounded domain with Lipschitz boundary, $M:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ and $g:\Omega \times \mathbb{R}\to \mathbb{R}$ are continuous functions, where $g$ behaves like $\exp (|u{|}^{\frac{n}{n-s}})$ as $|u|\to \infty$.

23 pages