### Abstract

Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain O ? R^{n}, n = 2, and a, b ? L^{8}(O). If the equation Lu = au^{+} - bu^{-} + ?u (where ? ? R and u^{±}(x) = max{±u(x),0}) has a non-trivial solution u, then ? is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are 'Simple'. We also consider the semilinear problem Lu = f(x, u), where f: O × R ? R is a Carathéodory function such that, for a.e. x ? O, a(x) = lim _{??8} f(x,?)/?, b(x) = lim _{??-8} f(x,?)/? and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).

Original language | English |
---|---|

Pages (from-to) | 1439-1451 |

Number of pages | 13 |

Journal | Proceedings of the Royal Society of Edinburgh, Section A: Mathematics |

Volume | 132 |

Issue number | 6 |

Publication status | Published - 2002 |

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### Cite this

*Proceedings of the Royal Society of Edinburgh, Section A: Mathematics*,

*132*(6), 1439-1451.

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*Proceedings of the Royal Society of Edinburgh, Section A: Mathematics*, vol. 132, no. 6, pp. 1439-1451.

**Half-eigenvalues of elliptic operators.** / Rynne, Bryan P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Half-eigenvalues of elliptic operators

AU - Rynne, Bryan P.

PY - 2002

Y1 - 2002

N2 - Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain O ? Rn, n = 2, and a, b ? L8(O). If the equation Lu = au+ - bu- + ?u (where ? ? R and u±(x) = max{±u(x),0}) has a non-trivial solution u, then ? is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are 'Simple'. We also consider the semilinear problem Lu = f(x, u), where f: O × R ? R is a Carathéodory function such that, for a.e. x ? O, a(x) = lim ??8 f(x,?)/?, b(x) = lim ??-8 f(x,?)/? and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).

AB - Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain O ? Rn, n = 2, and a, b ? L8(O). If the equation Lu = au+ - bu- + ?u (where ? ? R and u±(x) = max{±u(x),0}) has a non-trivial solution u, then ? is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are 'Simple'. We also consider the semilinear problem Lu = f(x, u), where f: O × R ? R is a Carathéodory function such that, for a.e. x ? O, a(x) = lim ??8 f(x,?)/?, b(x) = lim ??-8 f(x,?)/? and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).

UR - http://www.scopus.com/inward/record.url?scp=0036979601&partnerID=8YFLogxK

M3 - Article

VL - 132

SP - 1439

EP - 1451

JO - Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

SN - 0308-2105

IS - 6

ER -