In this work we study the nonlinear eigenvalue problem Lu = λ f(x , u) , x є D, with linear homogeneous boundary conditions, where D is a bounded domain in IRm with smooth boundary ∂D, and L is a uniformly elliptic, second order differential operator. The main objective of this work is to study the properties of the positive solutions and how they depend upon λ. We first show that the operator L, with some linear homogeneous boundary conditions is positive definite. Then we try to answer questions relating to the set of those λ (called the spectrum) for which positive solutions exist, their possible uniqueness. We also show that, if positive solutions exist, then the spectrum is an interval, say (0, λ*] or (0, λ*) (either possibility can occur), with 0 <λ* ≤ ∞. Furthermore, in this work we study the cases of concave and convex (with respect to u) functions f(x, u).