Journal of the Mathematical Society of Japan

Degenerate elliptic boundary value problems with asymmetric nonlinearity

Kazuaki TAIRA

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Abstract

This paper is devoted to the study of a class of semilinear degenerate elliptic boundary value problems with asymmetric nonlinearity which include as particular cases the Dirichlet and Robin problems. The most essential point is how to generalize the classical variational approach to eigenvalue problems with an indefinite weight to the degenerate case. The variational approach here is based on the theory of fractional powers of analytic semigroups. By making use of global inversion theorems with singularities between Banach spaces, we prove very exact results on the number of solutions of our problem. The results extend an earlier theorem due to Ambrosetti and Prodi to the degenerate case.

Article information

Source
J. Math. Soc. Japan Volume 62, Number 2 (2010), 431-465.

Dates
First available in Project Euclid: 7 May 2010

Permanent link to this document
http://projecteuclid.org/euclid.jmsj/1273236711

Digital Object Identifier
doi:10.2969/jmsj/06220431

Zentralblatt MATH identifier
05725850

Mathematical Reviews number (MathSciNet)
MR2662851

Subjects
Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35J25: Boundary value problems for second-order elliptic equations 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]

Keywords
semilinear elliptic boundary value problem degenerate boundary condition fractional power variational method global inversion theorem with singularities

Citation

TAIRA, Kazuaki. Degenerate elliptic boundary value problems with asymmetric nonlinearity. J. Math. Soc. Japan 62 (2010), no. 2, 431--465. doi:10.2969/jmsj/06220431. http://projecteuclid.org/euclid.jmsj/1273236711.


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