In this paper we consider the following nonhomogeneous semilinear fractional Laplacian problem $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u+u=\lambda (f(x,u)+h(x)) \,\, \text {in}\,\, \mathbb {R}^N,\\ u\in H^s(\mathbb {R}^N), u>0\,\, \text {in}\,\, \mathbb {R}^N, \end{array}\right. } \end{aligned}$$ where $$\lambda >0$$ and $$\lim _{|x|\rightarrow \infty }f(x,u)=\overline{f}(u)$$ uniformly on any compact subset of $$[0,\infty )$$ . We prove that under suitable conditions on f and h, there exists $$0<\lambda ^*<+\infty $$ such that the problem has at least two positive solutions if $$\lambda \in (0,\lambda ^*)$$ , a unique positive solution if $$\lambda =\lambda ^*$$ , and no solution if $$\lambda >\lambda ^*$$ . We also obtain the bifurcation of positive solutions for the problem at $$(\lambda ^*,u^*)$$ and further analyse the set of positive solutions.