Advances in Differential Equations

Positive solutions of semilinear elliptic eigenvalue problems with concave nonlinearities

Kenichiro Umezu

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In this paper, we prove the existence and nonexistence results for positive solutions to semilinear elliptic boundary value problems, with concave nonlinearities inside a smooth bounded domain and on the boundary. Our approach relies on sub and supersolutions, as well as the Nehari manifold that may contain the critical points for the energy functional associated with the boundary value problem. The fibering method helps us to study the properties of the Nehari manifold.

Article information

Adv. Differential Equations, Volume 12, Number 12 (2007), 1415-1436.

First available in Project Euclid: 18 December 2012

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Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35J20: Variational methods for second-order elliptic equations 35J25: Boundary value problems for second-order elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations 47J30: Variational methods [See also 58Exx] 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)


Umezu, Kenichiro. Positive solutions of semilinear elliptic eigenvalue problems with concave nonlinearities. Adv. Differential Equations 12 (2007), no. 12, 1415--1436.

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