Abstract
The exact multiplicity of positive solutions of the singular semilinear equation Δu + λ f (u) = 0 with Dirichlet boundary condition is studied. The nonlinearity f tends to infinity at zero, it is the linear combination of the functions u −α and u p with α, p > 0. The number λ is a positive parameter. The goal is to determine all possible bifurcation diagrams that can occur for different values of α and p. These kinds of equations have been widely studied on general domains, here we focus on the exact number of radial solutions on balls that can be investigated by the shooting method. It is shown how the so-called time-map can be introduced by the shooting method, and its properties determining the exact number of the positive solutions of the boundary value problem are studied. The exact multiplicity of positive solutions is given in the one-dimensional case, for which all the possible bifurcation diagrams are listed. In the higher dimensional case the bifurcation diagrams are not known for all values of α and p, in this case some open problems are collected.
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Simon, P.L. Exact Multiplicity of Positive Solutions for a Class of Singular Semilinear Equations. Differ Equ Dyn Syst 18, 415–429 (2010). https://doi.org/10.1007/s12591-010-0051-5
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DOI: https://doi.org/10.1007/s12591-010-0051-5