Existence and non-existence of positive solutions for a class of fractional Laplacian equations

We consider a class of fractional Laplacian equations including the following as a special case:$ (-\Delta)^{\frac{\alpha}{2}}u = \frac{u^p}{1+|x|^\beta}, \quad x\in \mathbb{R}^n, $where $ n\geq 3 $, $ \alpha\in(0, 2) $, $ p>0 $ and $ \beta\in \mathbb R $. Using the scaling spheres method of Dai and Qin [9] we prove a general nonexistence result which implies that the above equation has no positive solution when $ 0< p<\frac{n+\alpha-2\beta}{n-\alpha} $ and $ \beta\leq \alpha $. Using a global bifurcation result of Dancer [10], the fractional Laplacian version of Pohozaev's identity due to Ros-Oton and Serra [24] and a limiting argument, we show that the above equation has a continuum of positive radial solutions when $ p\geq \frac{n+\alpha}{n-\alpha} $ and $ \beta>0 $. Together with the existence result of Ma [19] for the range $ \frac{n+\alpha-2\beta}{n-\alpha}<p<\frac{n+\alpha}{n-\alpha} $ with $ \beta>0 $, these give a relatively complete picture for the existence of positive solutions of the above equation (except for the critical case $ p = \frac{n+\alpha-2\beta}{n-\alpha} $) when $ 0<\beta\leq \alpha $, although many questions remain to be answered.