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

$$ \left \{ \textstyle\begin{array}{l} \quad - M\left (\displaystyle \int _{\mathbb{R}^{n}}\int _{\mathbb{R}^{n}} \frac{|u(x)-u(y)|^{\frac{n}{s}}}{|x-y|^{2n}}~dxdy\right ) (-\Delta )^{s}_{n/s} u=\left (\displaystyle \int _{\Omega }\frac{G(y,u)}{|x-y|^{\mu }}~dy \right )g(x,u) \; \text{in}\; \Omega , \\ \quad \quad u =0\quad \text{in} \quad \mathbb{R}^{n} \setminus \Omega , \end{array}\displaystyle \right . $$

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 $|u|\rightarrow \infty$. The key feature of this article is the presence of Kirchhoff model along with convolution type nonlinearity having exponential growth which appears in several physical and biological models.