### B P Rynne

#### Abstract

Let $\Om$ be a bounded domain in $\R^n$, $n \ge 1$, with $C^2$ boundary $\pa \Om$, and consider the semilinear elliptic boundary value problem \begin{align*} L u &= \la a u + g(\cdot,u)u, \quad {\rm in}\ \Om,\\ u &= 0, \quad {\rm on}\ \pa\Om, \end{align*} where $L$ is a uniformly elliptic operator on $\bOm$, $a \in C^0(\bOm)$, $a$ is strictly positive in $\bOm$, and the function $g:\bOm \X \R \tends \R$ is continuously differentiable, with $g(x,0) = 0$, $x \in \bOm$. A well known result of Rabinowitz shows that an unbounded continuum of positive solutions of this problem bifurcates from the principal eigenvalue $\la_1$ of the linear problem. We show that under certain oscillation conditions on the nonlinearity $g$, this continuum oscillates about $\la_1$, in a certain sense, as it approaches infinity. Hence, in particular, the equation has infinitely many positive solutions for each $\la$ in an open interval containing $\la_1$.

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