Age and nature of 560–520 Ma calc-alkaline granitoids of Biarjmand, northeast Iran: insights into Cadomian arc magmatism in northern Gondwana

The Biarjmand granitoids and granitic gneisses in northeast Iran are part of the Torud–Biarjmand metamorphic complex, where previous zircon U–Pb geochronology show ages of ca. 554–530 Ma for orthogneissic rocks. Our new U–Pb zircon ages confirm a Cadomian age and show that the granitic gneiss is ~30 million years older (561.3 ± 4.7 Ma) than intruding granitoids(522.3 ± 4.2 Ma; 537.7 ± 4.7 Ma). Cadomian magmatism in Iran was part of an approximately 100-million-year-long episode of subduction-related arc and back-arc magmatism, which dominated the whole northern Gondwana margin, from Iberia to Turkey and Iran. Major REE and trace element data show that these granitoids have calc-alkaline signatures. Their zircon O (δ18O = 6.2–8.9‰) and Hf (–7.9 to +5.5; one point with εHf ~ –17.4) as well as bulk rock Nd isotopes (εNd(t)= –3 to –6.2) show that these magmas were generated via mixing of juvenile magmas with an older crust and/or melting of middle continental crust. Whole-rock Nd and zircon Hf model ages (1.3–1.6 Ga) suggest that this older continental crust was likely to have been Mesoproterozoic or even older. Our results, including variable zircon εHf(t) values, inheritance of old zircons and lack of evidence for juvenile Cadomian igneous rocks anywhere in Iran, suggest that the geotectonic setting during late Ediacaran and early Cambrian time was a continental magmatic arc rather than back-arc for the evolution of northeast Iran Cadomian igneous rocks.

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.