Nonlinear differential equations having non-sign-definite weights

Abstract

In the present PhD thesis we deal with the study of the existence, multiplicity and complex behaviors of solutions for some classes of boundary value problems associated with second order nonlinear ordinary differential equations of the form $u''+f(u)u'+g(t,u)=s,$ or $u''+g(t,u)=0,$ $t\in I$, where $I$ is a bounded interval, $f\colon\mathbb{R}\to\mathbb{R}$ is continuous, $s\in\mathbb{R}$ and $g: I\times \mathbb{R}\to\mathbb{R}$ is a perturbation term characterizing the problems. The results carried out in this dissertation are mainly based on dynamical and topological approaches. The issues we address have arisen in the field of partial differential equations. For this reason, we do not treat only the case of ordinary differential equations, but also we take advantage of some results achieved in the one dimensional setting to give applications to nonlinear boundary value problems associated with partial differential equations. In the first part of the thesis, we are interested on a problem suggested by Antonio Ambrosetti in ``Observations on global inversion theorems'' (2011). In more detail, we deal with a periodic boundary value problem associated with the first differential equation where the perturbation term is given by $g(t,u):=a(t)\phi(u)-p(t)$. We assume that $a,$ $p\in L^{\infty}(I)$ and $\phi\colon\mathbb{R}\to\mathbb{R}$ is a continuous function satisfying $\lim_{|\xi|\to\infty}\phi(\xi)=+\infty$. In this context, if the weight term $a(t)$ is such that $a(t)\geq 0$ for a.e. $t\in I$ and $\int_{I}a(t)\,dt>0$, we generalize the result of multiplicity of solutions given by Fabry, Mawhin and Nakashama in ``A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations'' (1986). We extend this kind of improvement also to more general nonlinear terms under local coercivity conditions. In this framework, we also treat in the same spirit Neumann problems associated with second order ordinary differential equations and periodic problems associated with first order ones. Furthermore, we face the classical case of a periodic Ambrosetti-Prodi problem with a weight term $a(t)$ which is constant and positive. Here, considering in the second differential equation a nonlinearity $g(t,u):=\phi(u)-h(t)$, we provide several conditions on the nonlinearity and the perturbative term that ensure the presence of complex behaviors for the solutions of the associated $T$-periodic problem. We also compare these outcomes with the result of stability carried out by Ortega in ``Stability of a periodic problem of Ambrosetti-Prodi type'' (1990). The case with damping term is discussed as well. In the second part of this work, we solve a conjecture by Yuan Lou and Thomas Nagylaki stated in ``A semilinear parabolic system for migration and selection in population genetics'' (2002). The problem refers to the number of positive solutions for Neumann boundary value problems associated with the second differential equation when the perturbation term is given by $g(t,u):=\lambda w(t)\psi(u)$ with $\lambda>0$, $w\in L^{\infty}(I)$ a sign-changing weight term such that $\int_{I}w(t)\,dt<0$ and $\psi\colon[0,1]\to[0,\infty[$ a non-concave continuous function satisfying $\psi(0)=0=\psi(1)$ and such that the map $\xi\mapsto \psi(\xi)/\xi$ is monotone decreasing. In addition to this outcome, other new results of multiplicity of positive solutions are presented as well, for both Neumann or Dirichlet boundary value problems, by means of a particular choice of indefinite weight terms $w(t)$ and different positive nonlinear terms $\psi(u)$ defined on the interval $[0,1]$ or on the positive real semi-axis $[0,+\infty[$