| Abstract: |
| We discuss existence of multiple positive solutions to some migration-selection models in population genetics governed by nonlinear differential equations of the form $u'' + q(t) g(u) = 0$, where $q(t)$ is sign-changing and $g\colon \mathopen{[}0,1\mathclose{]} \to \mathbb{R}$ is continuous with $g(0)=g(1)=0$, $g(s) > 0$ for $0 Inspired by the works of P.~H. Rabinowitz (Indiana Univ.~Math.~J., 1973/74), Y. Lou, W.-M. Ni, L. Su (Discrete Contin.~Dyn.~Syst., 2010), we obtain new multiplicity results dealing with various boundary conditions, including Dirichlet and Neumann ones. More exactly, using topological techniques based on shooting methods and on Mawhin's coincidence degree theory, we show how the number of positive solutions is affected by the nodal behaviour of $q(t)$.
We also focus on radially symmetric solutions to boundary value problems associated with $\Delta u + q(x)g(u)=0$.
This talk is based on joint works with A.~Boscaggin (University of Turin) and E.~Sovrano (University of Udine). |
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