Multiple positive solutions of nonhomogeneous elliptic equations in unbounded domains

We will show that under suitable conditions on f and h, there exists a positive number lambda* such that the nonhomogeneous elliptic equation -Delta u + u =lambda(f ( x, u) + h( x)) in Omega, u is an element of H(0)(1) (Omega), N >= 2, has at least two positive solutions if lambda is an element of (0,lambda*), a unique positive solution if lambda=lambda*, and no positive solution if lambda >lambda*, where Omega is the entire space or an exterior domain or an unbounded cylinder domain or the complement in a strip domain of a bounded domain. We also obtain some properties of the set of solutions.