Positive solutions of the Dirichlet problem for the prescribed mean curvature equation

Abstract

We discuss existence and multiplicity of positive solutions of the prescribed mean curvature problem − div ( ∇ u / 1 + | ∇ u | 2 ) = λ f ( x , u ) in Ω , u = 0 on ∂ Ω , in a general bounded domain Ω ⊂ R N , depending on the behavior at zero or at infinity of f ( x , s ) , or of its potential F ( x , s ) = ∫ 0 s f ( x , t ) d t . Our main effort here is to describe, in a way as exhaustive as possible, all configurations of the limits of F ( x , s ) / s 2 at zero and of F ( x , s ) / s at infinity, which yield the existence of one, two, three or infinitely many positive solutions. Either strong, or weak, or bounded variation solutions are considered. Our approach is variational and combines critical point theory, the lower and upper solutions method and elliptic regularization.