Bifurcation and multiplicity of positive solutions for nonhomogeneous fractional Schrodinger equations with critical growth

In this paper we study the nonhomogeneous semilinear fractional Schrodinger equation with critical growth {(-Delta)(s)u + u = u(s)(2)*(-1) + lambda(f(x, u) + h(x)), x is an element of R-N, u is an element of H-s(R-N), u(x) > 0, x is an element of R-N, where s is an element of (0, 1), N > 4s, and lambda > 0 is a parameter, 2*(s) = 2N/N-2s is the fractional critical Sobolev exponent, f and h are some given functions. We show that there exists 0 < lambda* < +infinity such that the problem has exactly two positive solutions if lambda is an element of (0, lambda*), no positive solutions for lambda > lambda*, a unique solution (lambda*,u(lambda*)) if lambda = lambda*, which shows that (lambda*,u(lambda*)) is a turning point in H-S (R-N) for the problem. Our proofs are based on the variational methods and the principle of concentration-compactness.