Exact multiplicity of solutions for some semilinear Dirichlet problems

TL;DR

This paper determines the exact number of solutions for certain semilinear Dirichlet problems depending on a parameter, extending classical results to more general functions and applications, including a new model and a simple proof of an anti-maximum principle.

Contribution

It generalizes classical solution multiplicity results to problems with positive forcing functions using generalized harmonics, and introduces a new model relevant to fishing applications.

Findings

Exact solution counts depending on parameter μ

Extension to problems with positive functions f(x)

Simplified proof of the anti-maximum principle

Abstract

The classical result of A. Ambrosetti and G. Prodi [1], in the form of M.S. Berger and E. Podolak [4], gives the exact number of solutions for the problem \[ \Delta u+g(u)= \mu \phi _1(x)+e(x) \;\; \mbox{in $D$} , \;\; u=0 \;\; \mbox{on $\partial D$} \,, \] depending on the real parameter $\mu$, for a class of convex $g(u)$, and $\int _D e(x) \phi _1(x)\, dx=0$ (where $\phi _1(x)>0$ is the principal eigenfunction of the Laplacian on $D$, and $D \subset R^n$ is a smooth domain). By considering generalized harmonics, we give a similar result for the problem \[ \Delta u+g(u)= \mu f(x) \;\; \mbox{in $D$} , \;\; u=0 \;\; \mbox{on $\partial D$} \,, \] with $f(x)>0$. Such problems occur, for example, in "fishing" applications that we discuss, and propose a new model. Our approach also produces a very simple proof of the anti-maximum principle of Ph. Cl\'{e}ment and L.A. Peletier [5].