Positive Solutions of an Indefinite Prescribed Mean Curvature Problem on a General Domain

Abstract

The existence of positive solutions is proved for the prescribed mean curvature problem where Ω ⊂ℝ^N is a bounded smooth domain, not necessarily radially symmetric. We assume that ∫₀ ^u f(x, s) ds is locally subquadratic at 0, ∫₀ ^u g(x, s) ds is superquadratic at 0 and λ > 0 is sufficiently small. A multiplicity result is also obtained, when ∫₀ ^u f(x, s) ds has an oscillatory behaviour near 0. We allow f and g to change sign in any neighbourhood of 0.