Multiplicity results for some classes of Schrödinger-Poisson systems

In this thesis, we study the existence and multiplicity of solutions of the following class of Schr odinger-Poisson systems: u + u + l(x) u = (x; u) in R3; = l(x)u2 in R3; where l 2 L2(R3) or l 2 L1(R3). And we consider that the nonlinearity satis es the following three kinds of cases: (i) a subcritical exponent with (x; u) = k(x)jujp 2u + h(x)u (4 p < 2 ) under an inde nite case; (ii) a general inde nite nonlinearity with (x; u) = k(x)g(u) + h(x)u; (iii) a critical growth exponent with (x; u) = k(x)juj2 2u + h(x)jujq 2u (2 q < 2 ). It is worth mentioning that the thesis contains three main innovations except overcoming several di culties, which are generated by the systems themselves. First, as an unknown referee said in his report, we are the rst authors concerning the existence of multiple positive solutions for Schr odinger- Poisson systems with an inde nite nonlinearity. Second, we nd an interesting phenomenon in Chapter 2 and Chapter 3 that we do not need the condition R R3 k(x)ep 1dx < 0 with an inde nite noncoercive case, where e1 is the rst eigenfunction of +id in H1(R3) with weight function h. A similar condition has been shown to be a su cient and necessary condition to the existence of positive solutions for semilinear elliptic equations with inde nite nonlinearity for a bounded domain (see e.g. Alama-Tarantello, Calc. Var. PDE 1 (1993), 439{475), or to be a su cient condition to the existence of positive solutions for semilinear elliptic equations with inde nite nonlinearity in RN (see e.g. Costa-Tehrani, Calc. Var. PDE 13 (2001), 159{189). Moreover, the process used in this case can be applied to study other aspects of the Schr odinger-Poisson systems and it gives a way to study the Kirchho system and quasilinear Schr odinger system. Finally, to get sign changing solutions in Chapter 5, we follow the spirit of Hirano-Shioji, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 333, but the procedure is simpler than that they have proposed in their paper.

NMR in the characterization of heavy residual procedural streams

The main objective of this work was to monitor a set of physical-chemical properties of heavy oil procedural streams through nuclear magnetic resonance spectroscopy, in order to propose an analysis procedure and online data processing for process control. Different statistical methods which allow to relate the results obtained by nuclear magnetic resonance spectroscopy with the results obtained by the conventional standard methods during the characterization of the different streams, have been implemented in order to develop models for predicting these same properties. The real-time knowledge of these physical-chemical properties of petroleum fractions is very important for enhancing refinery operations, ensuring technically, economically and environmentally proper refinery operations. The first part of this work involved the determination of many physical-chemical properties, at Matosinhos refinery, by following some standard methods important to evaluate and characterize light vacuum gas oil, heavy vacuum gas oil and fuel oil fractions. Kinematic viscosity, density, sulfur content, flash point, carbon residue, P-value and atmospheric and vacuum distillations were the properties analysed. Besides the analysis by using the standard methods, the same samples were analysed by nuclear magnetic resonance spectroscopy. The second part of this work was related to the application of multivariate statistical methods, which correlate the physical-chemical properties with the quantitative information acquired by nuclear magnetic resonance spectroscopy. Several methods were applied, including principal component analysis, principal component regression, partial least squares and artificial neural networks. Principal component analysis was used to reduce the number of predictive variables and to transform them into new variables, the principal components. These principal components were used as inputs of the principal component regression and artificial neural networks models. For the partial least squares model, the original data was used as input. Taking into account the performance of the develop models, by analysing selected statistical performance indexes, it was possible to conclude that principal component regression lead to worse performances. When applying the partial least squares and artificial neural networks models better results were achieved. However, it was with the artificial neural networks model that better predictions were obtained for almost of the properties analysed. With reference to the results obtained, it was possible to conclude that nuclear magnetic resonance spectroscopy combined with multivariate statistical methods can be used to predict physical-chemical properties of petroleum fractions. It has been shown that this technique can be considered a potential alternative to the conventional standard methods having obtained very promising results.