On the semilinear fractional elliptic equations with singular weight functions

Abstract

In the paper, we study a class of semilinear fractional semilinear elliptic equations involving concave-convex nonlinearities: \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{\alpha} u+V_{\lambda }\left( x\right) u = f\left( x\right) \left\vert u\right\vert ^{q-2}u+g\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{in }\mathbb{R}^{N}, u\in H^{\alpha}(\mathbb{R}^{N}), & \end{array}\right. \end{equation*} $\end{document} where \begin{document}$ \alpha\in (0,1] $\end{document} , \begin{document}$ 1 2\alpha\right), $\end{document} the potential \begin{document}$ V_{\lambda }(x) = \lambda a(x)-b(x) $\end{document} and the parameter \begin{document}$ \lambda >0. $\end{document} Under some suitable assumptions on \begin{document}$ a,b $\end{document} and the weight functions \begin{document}$ f,g $\end{document} , we obtain the existence and multiplicity of non-trivial (positive) solutions for \begin{document}$ \lambda $\end{document} large enough. An interesting phenomenon is that we do not need the condition that weight functions \begin{document}$ f, g $\end{document} are integrable or bounded on whole space \begin{document}$ \mathbb{R}^{N}. $\end{document}