On the Existence of a Non-trivial Non-negative Global Radial Weak Solution to a Fractional Laplacian Problem with a Singular Potential

We prove the existence of a non-trivial non-negative radial weak solution to the problem \begin{equation*} \begin{cases} (-\Delta) ^{\alpha} u+bu=\lambda \dfrac{u}{|x|^{2\alpha}} +|u|^{p-1}u+\mu |u|^{r-1}u & \mathrm{in} \ \mathbb{R}^N,\\ \lim\limits_{|x| \to \infty} u(x) = 0. \end{cases} \end{equation*} Here $N \gt 2\alpha $, $ \alpha \in ({1}/{2},1)$, $1 \lt r \lt p \lt \frac{N+2\alpha}{N-2\alpha}$ and $\mu \in \mathbb{R}$. We also assume that $b \gt 0$ and $ 0 \lt \lambda \lt 4^\alpha \frac{\Gamma^2(\frac{N+2\alpha}4)}{\Gamma^2(\frac{N-2\alpha}4)}$.