If X is a simply connected CW complex, then it has a unique (up to isomorphism) minimal Sullivan model. There is an important rational homotopy invariant, called the rational Lusternik–Schnirelmann of X, denoted cat0(X), which has an algebraic formulation in terms of the minimal Sullivan model of X. We study another such numerical invariant called the rational retraction index of X, denoted r0(X), which is defined in terms of the minimal Sullivan model of X and satisfies 0 ≤ r0(X) ≤ cat0(X). It was introduced by Cuvilliez et al. as a tool to estimate the rational Lusternik–Schnirelmann category of the total space of a fibration. In this thesis we compute the rational retraction index on a range of rationally elliptic spaces, including for example spheres, complex projective space, the biquotient Sp(1) \ Sp(3) / Sp(1) × Sp(1), the homogeneous space Sp(3)/U(3) and products of these. In particular, we focus on formal spaces and formulate a conjecture to answer a question posed in the original article of Cuvilliez et al., “If X is formal, what invariant of the algebra H∗(X;Q) is r0(X)?”

If X is a simply connected CW complex, then it has a unique (up to isomorphism) minimal Sullivan model. There is an important rational homotopy invariant, called the rational Lusternik–Schnirelmann of X, denoted cat0(X), which has an algebraic formulation in terms of the minimal Sullivan model of X. We study another such numerical invariant called the rational retraction index of X, denoted r0(X), which is defined in terms of the minimal Sullivan model of X and satisfies 0 ≤ r0(X) ≤ cat0(X). It was introduced by Cuvilliez et al. as a tool to estimate the rational Lusternik–Schnirelmann category of the total space of a fibration. In this thesis we compute the rational retraction index on a range of rationally elliptic spaces, including for example spheres, complex projective space, the biquotient Sp(1) \ Sp(3) / Sp(1) × Sp(1), the homogeneous space Sp(3)/U(3) and products of these. In particular, we focus on formal spaces and formulate a conjecture to answer a question posed in the original article of Cuvilliez et al., “If X is formal, what invariant of the algebra H∗(X;Q) is r0(X)?”

If X is a simply connected CW complex, then it has a unique (up to isomorphism) minimal Sullivan model. There is an important rational homotopy invariant, called the rational Lusternik–Schnirelmann of X, denoted cat0(X), which has an algebraic formulation in terms of the minimal Sullivan model of X. We study another such numerical invariant called the rational retraction index of X, denoted r0(X), which is defined in terms of the minimal Sullivan model of X and satisfies 0 ≤ r0(X) ≤ cat0(X). It was introduced by Cuvilliez et al. as a tool to estimate the rational Lusternik–Schnirelmann category of the total space of a fibration. In this thesis we compute the rational retraction index on a range of rationally elliptic spaces, including for example spheres, complex projective space, the biquotient Sp(1) \ Sp(3) / Sp(1) × Sp(1), the homogeneous space Sp(3)/U(3) and products of these. In particular, we focus on formal spaces and formulate a conjecture to answer a question posed in the original article of Cuvilliez et al., “If X is formal, what invariant of the algebra H∗(X;Q) is r0(X)?”

In this thesis we study Schrödinger-Poisson systems and we look for positive solutions. Our work consists in three chapters. Chapter 1 includes some basic facts on critical point theory. In Chapter 2 we consider a fractional Schrödinger-Poisson system in the whole space R^N in presence of a positive potential and depending on a small positive parameter . We show that, for suitably small (i.e. in the \"semiclassical limit\") the number of positive solutions is estimated below by the Ljusternick-Schnirelmann category of the set of minima of the potential. Finally, in Chapter 3, we analyze a Schrödinger-Poisson system in R^3 under an asymptotically cubic nonlinearity. We prove the existence of positive, radial solutions inside a ball and in an exterior domain. / Nesta tese nós estudamos sistemas de Schrödinger-Poisson e procuramos soluções positivas. Nosso trabalho consiste em três capítulos. O Capítulo 1 contém alguns fatos básicos sobre a teoria de pontos críticos. No Capítulo 2 nós consideramos um sistema fracionário de Schrödinger-Poisson em todo o espaço R^N em presença de um potencial positivo e que depende de um pequeno parâmetro positivo . Nós mostramos que, para suficentemente pequeno (i.e. no limite semiclássico) o número de soluções positivas é estimado por abaixo pela categoria de Ljusternick-Schnirelmann dos conjuntos onde o potencial é mínimo. Finalmente, no Capítulo 3 nós analisamos um sistema Schrödinger-Poisson em R^3 sob a não linearidade assintoticamente cúbica. Mostramos a existência de soluções radiais positivas dentro de uma bola e em um domínio exterior.

In this thesis we study Schrödinger-Poisson systems and we look for positive solutions. Our work consists in three chapters. Chapter 1 includes some basic facts on critical point theory. In Chapter 2 we consider a fractional Schrödinger-Poisson system in the whole space R^N in presence of a positive potential and depending on a small positive parameter . We show that, for suitably small (i.e. in the \"semiclassical limit\") the number of positive solutions is estimated below by the Ljusternick-Schnirelmann category of the set of minima of the potential. Finally, in Chapter 3, we analyze a Schrödinger-Poisson system in R^3 under an asymptotically cubic nonlinearity. We prove the existence of positive, radial solutions inside a ball and in an exterior domain. / Nesta tese nós estudamos sistemas de Schrödinger-Poisson e procuramos soluções positivas. Nosso trabalho consiste em três capítulos. O Capítulo 1 contém alguns fatos básicos sobre a teoria de pontos críticos. No Capítulo 2 nós consideramos um sistema fracionário de Schrödinger-Poisson em todo o espaço R^N em presença de um potencial positivo e que depende de um pequeno parâmetro positivo . Nós mostramos que, para suficentemente pequeno (i.e. no limite semiclássico) o número de soluções positivas é estimado por abaixo pela categoria de Ljusternick-Schnirelmann dos conjuntos onde o potencial é mínimo. Finalmente, no Capítulo 3 nós analisamos um sistema Schrödinger-Poisson em R^3 sob a não linearidade assintoticamente cúbica. Mostramos a existência de soluções radiais positivas dentro de uma bola e em um domínio exterior.