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Applicable Analysis
An International Journal
Volume 98, 2019 - Issue 8
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Articles

Exact multiplicity of solutions for some semilinear Dirichlet problems

Pages 1483-1495 | Received 06 Oct 2017, Accepted 13 Jan 2018, Published online: 31 Jan 2018
 

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 Δu+g(u)=μφ1(x)+e(x)inD,u=0onD,

depending on the real parameter μ, for a class of convex g(u). Here, De(x)φ1(x)dx=0 (where φ1(x)>0 is the principal eigenfunction of the Laplacian on D, and DRn is a smooth domain). By considering generalized harmonics, we give a similar result for the problem Δu+g(u)=μf(x)inD,u=0onD,

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].

AMS SUBJECT CLASSIFICATIONS:

Disclosure statement

No potential conflict of interest was reported by the authors.

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