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.

Phthalocyanines : interaction with carbon structures and as PDT agents

This dissertation describes the synthesis and characterization of different phthalocyanine (Pc) derivatives, as well as some porphyrins (Pors), for supramolecular interaction with different carbon nanostructures, to evaluate their potential application in electronic nanodevices. Likewise, it is also reported the preparation and biological evaluation of interesting phthalocyanine conjugates for cancer photodynamic therapy (PDT) and microorganisms photodynamic inactivation (PDI). The phthalonitrile precursors were prepared from commercial phthalonitriles by nucleophilic substitution of -NO2, -Cl, or -F groups, present in the phthalonitrile core, by thiol or pyridyl units. After the synthesis of these phthalonitriles, the corresponding Pcs were prepared by ciclotetramerization using a metallic salt as template at high temperatures. A second strategy involved the postfunctionalization of hexadecafluorophthalocyaninato zinc(II) through the adequate substituents of mercaptopyridine or cyclodextrin units on the macrocycle periphery. The different compounds were structurally characterized by diverse spectroscopic techniques, namely 1H, 13C and 19F nuclear magnetic resonance spectroscopies (attending the elemental composition of each structure); absorption and emission spectroscopy, and mass spectrometry. For the specific photophysical studies were also used electrochemical characterization, femtosecond and raman spectroscopy, transmission electron and atomic force microscopy. It was highlighted the noncovalent derivatisation of carbon nanostructures, mainly single wall carbon nanotubes (SWNT) and graphene nanosheets with the prepared Pc conjugates to study the photophysical properties of these supramolecular nanoassemblies. Also, from pyridyl-Pors and ruthenium phthalocyanines (RuPcs) were performed Por-RuPcs arrays via coordination chemistry. The results obtained of the novel supramolecular assemblies showed interesting electron donor-acceptor interactions and might be considered attractive candidates for nanotechnological devices. On the other hand, the amphiphilic phthalocyanine-cyclodextrin (Pc-CD) conjugates were tested in biological trials to assess their ability to inhibit UMUC- 3 human bladder cancer cells. The results obtained demonstrated that these photoactive conjugates are highly phototoxic against human bladder cancer cells and could be applied as promising PDT drugs.