Journal of the Mathematical Society of Japan

Semilinear degenerate elliptic boundary value problems via Morse 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. By making use of the Morse and Ljusternik–Schnirelman theories of critical points, we prove existence theorems of non-trivial solutions of our problem. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of semilinear elliptic boundary value problems with degenerate boundary conditions. The results here extend earlier theorems due to Ambrosetti–Lupo and Struwe to the degenerate case.

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J. Math. Soc. Japan, Volume 67, Number 1 (2015), 339-382.

First available in Project Euclid: 22 January 2015

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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 Morse theory Ljusternik–Schnirelman theory


TAIRA, Kazuaki. Semilinear degenerate elliptic boundary value problems via Morse theory. J. Math. Soc. Japan 67 (2015), no. 1, 339--382. doi:10.2969/jmsj/06710339.

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