Existence of solutions to fractional elliptic problem with nonlocal gradient term and lower order term

Let

$ \Omega\subset \mathbb{R}^N \ \left(3\leqslant N<\frac{2s(2-\theta)-4t}{\theta}\right) $

be a smooth bounded domain. For

$ 0<t<\frac{2s(2-\theta)-3\theta}{4}<s<1, $

we consider the following fractional partial differential equation:

$ \begin{align*} \begin{cases} \begin{aligned} (-\Delta)^s u(x) & = B\left|{\mathbb{D}}_{t}\left(u^{\frac{2-\theta}{2}}(x)\right)\right|^2 + \lambda f(x), && x\in\Omega, \\ u(x) &> 0, && x\in \Omega, \\ u(x) & = 0, && x\in \mathbb{R}^N\setminus\Omega, \end{aligned} \end{cases} \end{align*} $

where $ B > 0 $ is a constant, $ \theta\in\left(0, \frac{4s}{2s+3}\right) $, $ 0 < f\in L^m(\Omega) $, and $ \lambda > 0 $ is a real parameter. In addition, $ {\mathbb{D}}_{t}\left(u^{\frac{2-\theta}{2}}(x)\right) $ denotes a nonlocal gradient term. Problems involving local gradient terms and lower-order terms have been extensively investigated in the existing literature. Motivated by this, we focus on the corresponding problem with nonlocal gradient terms and lower-order terms in the present paper. Specifically, our aim is to analyze the influence of the lower-order term on the existence of solutions to the fractional Laplace problem. For $ 0 < \lambda \leqslant \lambda^* $, we prove the existence of solutions to this problem when $ m > \frac{N}{s} $.