Critical growth problems with singular nonlinearities on Carnot groups

We provide regularity, existence and non existence results for the semilinear subelliptic problem with critical growth −ΔGu=ψ^α|u|^(2∗(α)−2)u/d(ξ)^α+λu in ΩΩ, u=0 on ∂Ω, where ΔG is a sublaplacian on a Carnot group GG, 0<2, 2∗(α)=2(Q−α)/(Q−2), Ω is a bounded domain of G, d is the natural gauge associated with the fundamental solution of −ΔG on G and ψ:=|∇Gd|, ∇G being the subelliptic gradient associated to ΔG, λ is a real parameter

Correlation between the quenching of total GT+ strength and the increase of E2 strength

Relations between the total beta+ Gamow-Teller (GT+) strength and the E2 strength are further examined. It is found that in shell-model calculations for N=Z nuclei, in which changes in deformation are induced by varying the single-particle energies, the total GT+ or GT- strength decreases monotonically with increasing values of the B(E2) from the ground state to the first excited J=2+ state. Similar trends are also seen for the double GT transition amplitude (with some exceptions) and for the spin part of the total M1 strength as a function of B(E2).Comment: 11 pages and 3 figures (Figures will be sent on request

The Effects of Deformation on Isovector Electromagnetic and Weak Transition Strengths

The summed strength for transitions from the ground state of $^{12}C$ via the operators $\vec{s}t, \vec{\ell}t, rY't, r[Y's]^{\lambda}t$ and $r[Y'\ell]^{\lambda}t$ are calculated using the $\Delta N = 0$ rotational model. If we choose the z component of the isospin operator $t_{z}$, the above operators are relevant to electromagnetic transitions; if we choose $t_{+}$ they are relevant to weak transitions such as neutrino capture. In going from the spherical limit to the asymptotic (oblate) limit the strength for the operator $\vec{s} t$ decreases steadily to zero; the strength for the operator $\vec{\ell}\tau$ (scissors mode) increases by a factor of three. For the last three operators - isovector dipole, spin dipole and orbital dipole (including the twist mode) it is shown that the summed strength is independant of deformation. The main difference in the behavior is that for the first two operators we have in-shell transitions whereas for the last three operators the transitions are out of shell.Comment: 14 pages, late