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

Abstract

<p style='text-indent:20px;'>In this paper, we consider the following general pseudo-relativistic Schrödinger equation with indefinite nonlinearities:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (-\Delta+m^{2})^{s}u = a(x_1)f\left(u,\nabla u\right),\quad {\rm{in}} \,\,\mathbb R^{N}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ s\in(0,1) $\end{document}</tex-math></inline-formula>, mass <inline-formula><tex-math id="M2">\begin{document}$ m>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ a $\end{document}</tex-math></inline-formula> 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.</p>