Monotonicity and nonexistence of positive solutions for pseudo-relativistic equation with indefinite nonlinearity

In this paper, we consider the following general pseudo-relativistic Schrödinger equation with indefinite nonlinearities:

\begin{document}$ (-\Delta+m^{2})^{s}u = a(x_1)f\left(u,\nabla u\right),\quad {\rm{in}} \,\,\mathbb R^{N}, $\end{document}

where \begin{document}$ s\in(0,1) $\end{document}, mass \begin{document}$ m>0 $\end{document} and \begin{document}$ a $\end{document} is a non-decreasing functions. We prove the nonexistence and the monotonicity of the positive bounded solution for the above equation via the direct method of moving planes.