Oscillating global continua of positive solutions of semilinear elliptic problems

Let Omega be a bounded domain in R-n, n greater than or equal to 1, with C-2 boundary partial derivative Omega, and consider the semilinear elliptic boundary value problem Lu = lambda au + g(., u)u, in Omega, u = 0, on partial derivative Omega, where L is a uniformly elliptic operator on <(Omega)over bar>, a is an element of C-0 (<(Omega)over bar>), a is strictly positive in <(Omega)over bar>, and the function g : <(Omega)over bar> x R --> R is continuously differentiable, with g(x; 0) = 0, x is an element of <(Omega)over bar>. A well known result of Rabinowitz shows that an unbounded continuum of positive solutions of this problem bifurcates from the principal eigenvalue lambda(1) of the linear problem. We show that under certain oscillation conditions on the nonlinearity g, this continuum oscillates about lambda(1), in a certain sense, as it approaches infinity. Hence, in particular, the equation has infinitely many positive solutions for each lambda in an open interval containing lambda(1).