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.
Kazuaki TAIRA. "Degenerate elliptic boundary value problems with asymmetric nonlinearity." J. Math. Soc. Japan 62 (2) 431 - 465, April, 2010. https://doi.org/10.2969/jmsj/06220431