### 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 |
---|---|

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 |

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### Keywords

- Global bifurcation
- Semilinear elliptic equations

### Cite this

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*Proceedings of the American Mathematical Society*, vol. 128, no. 1, pp. 229-236.

**Oscillating global continua of positive solutions of semilinear elliptic problems.** / Rynne, Bryan P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Oscillating global continua of positive solutions of semilinear elliptic problems

AU - Rynne, Bryan P.

PY - 2000

Y1 - 2000

N2 - 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.

AB - 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.

KW - Global bifurcation

KW - Semilinear elliptic equations

UR - http://www.scopus.com/inward/record.url?scp=22844455810&partnerID=8YFLogxK

M3 - Article

VL - 128

SP - 229

EP - 236

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 1

ER -