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

Let

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

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:

$$\begin{array}{r}\{\begin{array}{l}\begin{array}{rlrl}(-\mathrm{\Delta }{)}^{s}u(x)& =B{\left|{\mathbb{D}}_t\left(u^{\frac{2-\theta }{2}}(x)\right)\right|}^2+\lambda f(x),& & x\in \mathrm{\Omega },\\ u(x)& >0,& & x\in \mathrm{\Omega },\\ u(x)& =0,& & x\in {\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}\end{array}\end{array}$$

where $B>0$ is a constant, $\theta \in \left(0,\frac{4s}{2s+3}\right)$, $0<f\in L^m(\mathrm{\Omega })$, 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 $0<\lambda ⩽{\lambda }^{\ast }$, we prove the existence of solutions to this problem when $m>\frac{N}{s}$.