Journal of the Mathematical Society of Japan

Degenerate elliptic boundary value problems with asymmetric nonlinearity

Kazuaki TAIRA

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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

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

First available in Project Euclid: 7 May 2010

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Zentralblatt MATH identifier

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]

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


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.

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