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 b oundary 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 elem ent 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 differen tiable, with g(x; 0) = 0, x is an element of <(Omega)over bar>. A well know n result of Rabinowitz shows that an unbounded continuum of positive soluti ons of this problem bifurcates from the principal eigenvalue lambda(1) of t he linear problem. We show that under certain oscillation conditions on the nonlinearity g, this continuum oscillates about lambda(1), in a certain se nse, as it approaches infinity. Hence, in particular, the equation has infi nitely many positive solutions for each lambda in an open interval containi ng lambda(1).