Mathematics

The present Ph.D. thesis is devoted to the study of positive solutions to indefinite problems.
In particular, we deal with the second order nonlinear differential equation u'' + a(t) g(u) = 0,
where g : [0,+∞[→[0,+∞[ is a continuous nonlinearity and a : [0,T]→R is a Lebesgue integrable sign-changing weight.
We analyze the Dirichlet, Neumann and periodic boundary value problems on [0,T] associated with the equation and we provide existence, nonexistence and multiplicity results for positive solutions.

In the first part of the manuscript, we investigate nonlinearities g(u) with a superlinear growth at zero and at infinity (including the classical superlinear case g(u)=u^p, with p>1).
In particular, we prove that there exist 2^m-1 positive solutions when a(t) has m positive humps separated by negative ones and the negative part of a(t) is sufficiently large.
Then, for the Dirichlet problem, we solve a conjecture by Gómez‐Reñasco and López‐Gómez (JDE, 2000) and, for the periodic problem, we give a complete answer to a question raised by Butler (JDE, 1976).

In the second part, we study the super-sublinear case (i.e. g(u) is superlinear at zero and
sublinear at infinity). If a(t) has m positive humps separated by negative ones, we obtain the existence of 3^m-1 positive solutions of the boundary value problems associated with the parameter-dependent equation u'' + λ a(t) g(u) = 0, when both λ>0 and the negative part of a(t) are sufficiently large.

We propose a new approach based on topological degree theory for locally compact operators on open possibly unbounded sets, which applies for Dirichlet, Neumann and periodic boundary conditions.
As a byproduct of our method, we obtain infinitely many subharmonic solutions and globally defined positive solutions with complex behavior, and we deal with chaotic dynamics.
Moreover, we study positive radially symmetric solutions to the Dirichlet and Neumann problems associated with elliptic PDEs on annular domains.
Furthermore, this innovative technique has the potential and the generality needed to deal with indefinite problems with more general differential operators. Indeed, our approach apply also for the non-Hamiltonian equation u'' + cu' + a(t) g(u) = 0. Meanwhile, more general operators in the one-dimensional case and problems involving PDEs will be subjects of future investigations.

Mathematics

MAT/05