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Oscillating global continua of positive solutions of semilinear elliptic problems

Authors:	Bryan P Rynne

Institution:	Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland

Abstract:	Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ , $n \ge 1$ , with boundary $\partial \Omega$ , and consider the semilinear elliptic boundary value problem $\begin{align}L u &= \lambda a u + g(\cdot,u)u, \quad \text{in} \Omega , u &= 0, \quad \text{on} \partial \Omega , \end{align}$ where is a uniformly elliptic operator on $\overline{\Omega }$ , $a \in C^0(\overline{\Omega })$ , is strictly positive in $\overline{\Omega }$ , and the function $g:\overline{\Omega }\times \mathbb{R}\rightarrow \mathbb{R}$ is continuously differentiable, with , $x \in \overline{\Omega }$ . 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 , 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 $\lambda _1$ .

Keywords:	Global bifurcation semilinear elliptic equations

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