Kirchhoff equations with Choquard exponential type nonlinearity involving the fractional Laplacian

TL;DR

This paper establishes the existence of non-negative solutions for a nonlocal fractional Kirchhoff problem involving Choquard exponential nonlinearity, extending the analysis to fractional Laplacian operators with critical exponential growth.

Contribution

It introduces new existence results for fractional Kirchhoff equations with Choquard exponential nonlinearity, a novel combination in the study of nonlocal PDEs.

Findings

Proved existence of solutions under exponential growth conditions.

Extended fractional Kirchhoff theory to include Choquard-type nonlinearities.

Analyzed the problem with fractional Laplacian operators in bounded domains.

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 $|u|\rightarrow\infty$.