Positive solutions to indefinite problems: a topological approach

In this seminar, we present some recent existence and multiplicity results for positive solutions of boundary value problems associated with second-order nonlinear indefinite differential equations. More precisely, we deal with the ordinary differential equation

u?? + a(t)g(u) = 0,

where a: [0,T] ? R is a Lebesgue integrable sign-changing weight and g: [0,+?[ ? [0,+?[ is a continuous nonlinearity.
We focus on the periodic boundary value problem and on functions g(u) with superlinear growth at zero and at infinity (including the classical superlinear case g(u) = up, with p > 1). Exploiting a new approach based on topological degree theory, we show that there exist 2m ? 1 positive solutions when a(t) has m positive humps separated by negative ones and the negative part of a(t) is sufficiently large. In this manner, we give a complete answer to a question raised by Butler (JDE, 1976) and we solve a conjecture by G ?omez-Ren ?asco and L ?opez-G ?omez (JDE, 2000). The method also applies to Neumann and Dirichlet boundary conditions and, furthermore, provides a topological approach to detect infinitely many subharmonic solutions and globally defined positive solutions with chaotic behaviour.
Thereafter, we illustrate other directions for the research on indefinite problems: super-sublinear problems, models in population genetics, and also problems involving more general differential oper- ators, as the Minkowski-curvature one or the one-dimensional p-Laplacian. Exact multiplicity results and indefinite problems in the PDE setting are also discussed.
The talk is based on joint works with Alberto Boscaggin (University of Torino), Elisa Sovrano (University of Porto) and Fabio Zanolin (University of Udine) and on the book “Positive Solutions to Indefinite Problems. A Topological Approach” (Frontiers in Mathematics, Birkh ?auser/Springer, 2018).