Oscillating global continua of positive solutions of semilinear elliptic problems

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$, $n \ge 1$, with $C^2$ boundary $\partial \Omega$, and consider the semilinear elliptic boundary value problem

where $L$ is a uniformly elliptic operator on $\overline{\Omega }$, $a \in C^0(\overline{\Omega })$, $a$ is strictly positive in $\overline{\Omega }$, and the function $g:\overline{\Omega }\times \mathbb{R}\rightarrow \mathbb{R}$ is continuously differentiable, with $g(x,0) = 0$, $x \in \overline{\Omega }$. 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 $\lambda _1$.