### Abstract

The systems considered have the form {black small square} in Ω, {black small square} on ∂Ω, where {black small square} is a bounded domain, A is a matrix of second order elliptic operators, and γ is a real parameter. For simplicity the results are stated for a single equation, but the range of validity for systems is discussed. The first type of a priori estimates give lower bounds for sup {black small square} in terms of γ and |Ω| when {black small square} is superlinear, upper bounds for sup {black small square} when {black small square} is sublinear, and lower bounds for γ when {black small square} has linear growth. The second type of estimates generalize to systems results of Brezis and Turner and P. L. Lions for single equations; they give upper bounds for sup {black small square} in the superlinear case. Those estimates require A to be diagonal. None of the results require a variational structure for the system.

Original language | English (US) |
---|---|

Pages (from-to) | 123-129 |

Number of pages | 7 |

Journal | North-Holland Mathematics Studies |

Volume | 92 |

Issue number | C |

DOIs | |

State | Published - 1984 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**A Priori Estimates in Nonlinear Eigenvalue Problems for Elliptic Systems.** / Cosner, George.

Research output: Contribution to journal › Article

*North-Holland Mathematics Studies*, vol. 92, no. C, pp. 123-129. https://doi.org/10.1016/S0304-0208(08)73687-9

}

TY - JOUR

T1 - A Priori Estimates in Nonlinear Eigenvalue Problems for Elliptic Systems

AU - Cosner, George

PY - 1984

Y1 - 1984

N2 - The systems considered have the form {black small square} in Ω, {black small square} on ∂Ω, where {black small square} is a bounded domain, A is a matrix of second order elliptic operators, and γ is a real parameter. For simplicity the results are stated for a single equation, but the range of validity for systems is discussed. The first type of a priori estimates give lower bounds for sup {black small square} in terms of γ and |Ω| when {black small square} is superlinear, upper bounds for sup {black small square} when {black small square} is sublinear, and lower bounds for γ when {black small square} has linear growth. The second type of estimates generalize to systems results of Brezis and Turner and P. L. Lions for single equations; they give upper bounds for sup {black small square} in the superlinear case. Those estimates require A to be diagonal. None of the results require a variational structure for the system.

AB - The systems considered have the form {black small square} in Ω, {black small square} on ∂Ω, where {black small square} is a bounded domain, A is a matrix of second order elliptic operators, and γ is a real parameter. For simplicity the results are stated for a single equation, but the range of validity for systems is discussed. The first type of a priori estimates give lower bounds for sup {black small square} in terms of γ and |Ω| when {black small square} is superlinear, upper bounds for sup {black small square} when {black small square} is sublinear, and lower bounds for γ when {black small square} has linear growth. The second type of estimates generalize to systems results of Brezis and Turner and P. L. Lions for single equations; they give upper bounds for sup {black small square} in the superlinear case. Those estimates require A to be diagonal. None of the results require a variational structure for the system.

UR - http://www.scopus.com/inward/record.url?scp=77957301191&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957301191&partnerID=8YFLogxK

U2 - 10.1016/S0304-0208(08)73687-9

DO - 10.1016/S0304-0208(08)73687-9

M3 - Article

AN - SCOPUS:77957301191

VL - 92

SP - 123

EP - 129

JO - North-Holland Mathematics Studies

JF - North-Holland Mathematics Studies

SN - 0304-0208

IS - C

ER -