Exact multiplicity of solutions for some semilinear Dirichlet problems

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 $$\mathrm{\Delta }u+g(u)=\mu {\phi }_1(x)+e(x)\text{in}D,u=0\text{on}\partial D,$$

depending on the real parameter $\mu$, for a class of convex g(u). Here, ${\int }_{D}e(x){\phi }_1(x)\mathrm{d}x=0$ (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 $$\mathrm{\Delta }u+g(u)=\mu f(x)\text{in}D,u=0\text{on}\partial D,$$

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].

KEYWORDS:

• Global solution curves
• exact number of solutions
• the anti-maximum principle

AMS SUBJECT CLASSIFICATIONS:

• 35J61
• 35J25
• 92D25

Disclosure statement

No potential conflict of interest was reported by the authors.