Oscillating global continua of positive solutions of semilinear elliptic problems

Let O be a bounded domain in Rⁿ, n = 1, with C² boundary ?O, and consider the semilinear elliptic boundary value problem Lu = ?au + g(·, u)u, in O, u = 0, on ?O, where L is a uniformly elliptic operator on fi, O a ? C⁰(O¯), a is strictly positive in O¯, and the function g : O¯ × R ? R is continuously differentiable, with g(x, 0) = 0, x ? O¯. A well known result of Rabinowitz shows that an unbounded continuum of positive solutions of this problem bifurcates from the principal eigenvalue ?₁ of the linear problem. We show that under certain oscillation conditions on the nonlincarity g, this continuum oscillates about ?₁, in a certain sense, as it approaches infinity. Hence, in particular, the equation has infinitely many positive solutions for each A in an open interval containing ?₁. © 1099 American Mathematical Society.