Exact multiplicity of solutions for some semilinear Dirichlet problems

Abstract

ABSTRACTThe classical result of Ambrosetti and Prodi [1], in the form of Berger and Podolak [3], gives the exact number of solutions for the problem depending on the real parameter , for a class of convex g(u). Here, (where is the principal eigenfunction of the Laplacian on D, and is a smooth domain). By considering generalized harmonics, we give a similar result for the problem with . Such problems occur, for example, in ‘fishing’ applications that we discuss, and propose a new model with sign-changing solutions. Our approach also produces a very simple proof of the anti-maximum principle of Clément and Peletier [4].