On the existence threshold for positive solutions of p-laplacian equations with a concave-convex nonlinearity

We study the following boundary value problem with a concave-convex nonlinearity: \begin{equation*} \left\{ \begin{array}{r c l l} -\Delta_p u & = & \Lambda\,u^{q-1}+ u^{r-1} & \textrm{in }\Omega, \\ u & = & 0 & \textrm{on }\partial\Omega. \end{array}\right. \end{equation*} Here $\Omega \subset \mathbb{R}^n$ is a bounded domain and $1<q<p<r<p^*$. It is well known that there exists a number $\Lambda_{q,r}>0$ such that the problem admits at least two positive solutions for $0<\Lambda<\Lambda_{q,r}$, at least one positive solution for $\Lambda=\Lambda_{q,r}$, and no positive solution for $\Lambda > \Lambda_{q,r}$. We show that $$\lim_{q \to p} \Lambda_{q,r} = \lambda_1(p),$$ where $\lambda_1(p)$ is the first eigenvalue of the p-laplacian. It is worth noticing that $\lambda_1(p)$ is the threshold for existence/nonexistence of positive solutions to the above problem in the limit case $q=p$

Uniqueness and existence of solutions in the BVt(Q) space to a doubly nonlinear parabolic problem

In this paper we present some results on the uniqueness and existence of a class of weak solutions (the so called BV solutions) of the Cauchy-Dirichlet problem associated to the doubly nonlinear diffusion equation b(u)_t-\operatorname{div}(|\nabla u-k(b(u))\boldkey{e}|^{p-2}(\nabla u-k(b(u))\boldkey{e})) +g(x,u) = f(t,x). This problem arises in the study of some turbulent regimes: flows of incompressible turbulent fluids through porous media, gases flowing in pipelines, etc. The solvability of this problem is established in the $BV_t(Q)$ space. We prove some comparison properties (implying uniqueness) when the set of jumping points of the BV solution has $N$-dimensional null measure and suitable additional conditions as, for instance, $b^{-1}$ locally Lipschitz. The existence of this type of weak solution is based on suitable uniform estimates of the BV norm of an approximated solution

Understanding chemical evolution in resolved galaxies -- I The local star fraction-metallicity relation

This work studies the relation between gas-phase oxygen abundance and stellar-to-gas fraction in nearby galaxies. We first derive the theoretical prediction, and argue that this relation is fundamental, in the sense that it must be verified regardless of the details of the gas accretion and star formation histories. Moreover, it should hold on "local" scales, i.e. in regions of the order of 1 kpc. These predictions are then compared with a set of spectroscopic observations, including both integrated and resolved data. Although the results depend somewhat on the adopted metallicity calibration, observed galaxies are consistent with the predicted relation, imposing tight constraints on the mass-loading factor of (enriched) galactic winds. The proposed parametrization of the star fraction-metallicity relation is able to describe the observed dependence of the oxygen abundance on gas mass at fixed stellar mass. However, the "local" mass-metallicity relation also depends on the relation between stellar and gas surface densities.Comment: 10 pages, 4 figures. Matches accepted version (significant typo corrected

Astrophysical parameters and orbital solution of the peculiar X-ray transient IGR J00370+6122

BD+6073 is the optical counterpart of the X-ray source IGR J00370+6122, a probable accretion-powered X-ray pulsar. The X-ray light curve of this binary system shows clear periodicity at 15.7 d, which has been interpreted as repeated outbursts around the periastron of an eccentric orbit. We obtained high-resolution spectra of BD+6073 at different epochs. We used the FASTWind code to generate a stellar atmosphere model to fit the observed spectrum and obtain physical magnitudes. The synthetic spectrum was used as a template for cross-correlation with the observed spectra to measure radial velocities. The radial velocity curve provided an orbital solution for the system. We have also analysed the RXTE/ASM and Swift/BAT light curves to confirm the stability of the periodicity. BD +6073 is a BN0.7 Ib low-luminosity supergiant located at an approximate distance of 3.1 kpc, in the CasOB4 association. We derive Teff=24000 K and log gc=3.0, and chemical abundances consistent with a moderately high level of evolution. The spectroscopic and evolutionary masses are consistent at the 1 sigma level with a mass of 15 solar masses. The recurrence time of the X-ray flares is the orbital period of the system. The NS is in a high eccentricity (e=0.56) orbit, and the X-ray emission is strongly peaked around orbital phase 0.2, though the observations are consistent with some level of X-ray activity happening at all orbital phases. The X-ray behaviour of IGR J00370+6122 is reminiscent of intermediate SFXTs, though its peak luminosity is rather low. The orbit is somewhat wider than those of classical persistent supergiant X-ray binaries, which, combined with the low luminosity of the mass donor, explains the low X-ray luminosity. IGR J00370+6122 will likely evolve towards a persistent supergiant system, highlighting the evolutionary connection between different classes of wind-accreting X-ray sources.Comment: Accepted for publication in A&