Existence of positive solutions for a general nonlinear eigenvalue problem

Let Omega subset of (R)n be a bounded domain, H = L-2(Omega), L : D( L) subset of H -> H be an unbounded linear operator, f is an element of C((Omega) over bar x R, R) and lambda is an element of R. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem Lu = lambda f( x, u), u is an element of D(L), which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a second-order ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.