On eigenvalue problems with Robin type boundary conditions having indefinite coefficients

Abstract

A Robin type boundary condition with a sign-changing coefficient is treated. First, the associated linear elliptic eigenvalue problem is studied, where the existence of a principal eigenvalue is discussed by the use of a variational approach. Second, the associated semilinear elliptic boundary value problem of logistic type is studied and the one parameter-dependent structure of positive solutions is investigated, where results obtained are due to the construction of suitable super- and subsolutions by using the principal positive eigenfunctions of the linear eigenvalue problem.

Keywords:

• Eigenvalue problem
• Indefinite Robin boundary condition
• Principal eigenvalue
• Variational characterization
• Lower estimate
• Population dynamics

Keywords:

• AMS 2000 Mathematics Subject Classifications:
• 35J65
• 35J20
• 35P30
• 35B32
• 92D25

Acknowledgements

The author would like to thank the referee for valuable comments and suggestions. This research was partly supported by the Grant-in-Aid for Scientific Research (C) (2), No. 16540165, Japan Society for the Promotion of Science.