Multiplicity and bifurcation of positive solutions for nonhomogeneous semilinear fractional Laplacian problems

In this paper we consider the following nonhomogeneous semilinear fractional Laplacian problem { (I-Delta)(s) u+u = lambda (f (x, u) +h(x)) in R-N, u is an element of H-s (R-N), u > 0 in R-N, where lambda > 0 and lim (|x|) -> infinity f (x, u) = (f) over bar (u) uniformly on any compact subset of [0, infinity). We prove that under suitable conditions on f and h, there exists 0 < lambda* < + infinity such that the problem has at least two positive solutions if lambda is an element of (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.