Exact multiplicity of solutions for some semilinear Dirichlet problems. Issue 8 (11th June 2019) - Korman, Philip
ABSTRACT: The classical result of Ambrosetti and Prodi [1], in the form of Berger and Podolak [3], gives the exact number of solutions for the problem Δ u + g ( u ) = μ φ 1 ( x ) + e ( x ) in D, u = 0 on ∂ D, depending on the real parameter μ, for a class of convex g ( u ). Here, ∫ D e ( x ) φ 1 ( x ) d x = 0 (where φ 1 ( x ) > 0 is the principal eigenfunction of the Laplacian on D, and D ⊂ R n is a smooth domain). By considering generalized harmonics, we give a similar result for the problem Δ u + g ( u ) = μ f ( x ) in D, u = 0 on ∂ D, with f ( x ) > 0 . 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].