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.