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On semilinear elliptic equations with indefinite nonlinearities. (English) Zbl 0809.35022
Summary: This paper concerns semilinear elliptic equations whose nonlinear term has the form \(W(x) f(u)\) where \(W\) changes sign. We study the existence of positive solutions and their multiplicity. The important role played by the negative part of \(W\) is contained in a condition which is shown to be necessary for homogeneous \(f\). More general existence questions are also discussed.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B32 Bifurcations in context of PDEs
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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