### Abstract

Let O be a bounded domain in R^{n}, n = 1, with C^{2} 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 ? C^{0}(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.

Original language | English |
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Pages (from-to) | 229-236 |

Number of pages | 8 |

Journal | Proceedings of the American Mathematical Society |

Volume | 128 |

Issue number | 1 |

Publication status | Published - 2000 |

### Keywords

- Global bifurcation
- Semilinear elliptic equations