Let \(\Omega \subset \mathbb {R}^N(N\geqslant 3)\) be a smooth bounded domain. For \(\theta \in \left( 0,1\right) \) and \(s\in \left( \frac{1}{2},\frac{2-\theta }{2}\right) \) , we establish the existence of solutions to the following fractional elliptic problem with nonlocal quadratic gradient terms by Schauder fixed point theorem \(\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} (-\Delta )^s u(x)& =\mu (x){{\mathcal {\mathbb {D}}^{2}_{s}\left( u^{\frac{2-\theta }{2}}(x)\right) }} + \lambda f(x),& & x\in \Omega ,\\ u(x)& >0,& & x\in \Omega ,\\ u(x)& =0,& & x\in {\mathbb {R}^N}\setminus {\Omega }, \end{aligned} \end{array}\right. } \end{aligned}\) where \(0<\mu (x)\in L^\infty (\Omega )\) , \(\lambda >0\) is a real parameter, \(0<f(x)\in L^m(\Omega )\) with \(m>\frac{N}{2s}\) . \({\mathcal {\mathbb {D}}^{2}_{s}\left( \cdot \right) }\) is a nonlocal quadratic gradient term. This problem corresponds to classical Laplace equations with quadratic gradient and singular lower order terms. The main results of this paper show that the mild singularities at zero (since \(\theta \in \left( 0,1\right) \) ) have a significant impact on the range of the fractional exponent s, which is a property unique to nonlocal operators.
