Tsukuba Journal of Mathematics

Semilinear degenerate elliptic boundary value problems via critical point theory

Kazuaki Taira

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The purpose of this paper is to study a class of semilinear elliptic boundary value problems with degenerate boundary conditions which include as particular cases the Dirichlet and Robin problems. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. By making use of a variant of the Ljusternik-Schnirelman theory of critical points, we prove very exact results on the number of solutions of our problem. The results here extend earlier theorems due to Castro-Lazer to the degenerate case.

Article information

Tsukuba J. Math. Volume 36, Number 2 (2013), 311-365.

First available in Project Euclid: 21 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35J20: Variational methods for second-order elliptic equations 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)

Semilinear elliptic boundary value problem degenerate boundary condition multiple solution critical point theory Ljusternik-Schnirelman theory


Taira, Kazuaki. Semilinear degenerate elliptic boundary value problems via critical point theory. Tsukuba J. Math. 36 (2013), no. 2, 311--365. doi:10.21099/tkbjm/1358777003. https://projecteuclid.org/euclid.tkbjm/1358777003

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