Multiple positive solutions for a p-Laplace critical problem (p >1), via Morse theory

summary:We consider the quasilinear elliptic problem $$(P_\lambda) \ \ \left\{ \begin{array}{ll} - \Delta _{p}u = \lambda u^{q-1} + u^{p^*-1} & \hbox{in} \ \Omega \\ u>0 & \hbox{in} \ \Omega \\ u=0 & \hbox{on} \ \partial \Omega \end{array} \right.$$ where $\Omega$ is bounded in $\R^N$, $N \geq p^2, \, 10$ is a parameter. Denoting by {\mathcal P}_1(\O) the Poincar\´e polynomial of $0$, we state that, for any $p>1$, there exists $\lambda^*>0$ such that, for any $\lambda\in (0,\lambda^*)$, either $(P_\lambda)$ has at least {\mathcal P}_1(\O) distinct solutions or, if not, (P_\l) can be approached by a sequence of problems $(P_n)_{n \in \N}$, each having at least $(P_n)_{n \in \N}$ distinct solutions. These results have been proved in [12] only as regards the case p 2, while they will be completely proved in the forthcoming work [13] in the case $p\geq2$, while they will be completely proved in the forthcoming work \cite{cvip} in the case $p\in (1,2)$. Note that, when $p\geq2$, the Euler functional associated to (P_\l) is never $C^2$, so the approach already used for $p\geq 2$ fails. This problem will be faced exploiting recent results given in [7] and [8]

Viruses as co-factors for the initiation or exacerbation of lung fibrosis

Idiopathic pulmonary fibrosis (IPF) remains exactly that. The disease originates from an unknown cause, and little is known about the mechanisms of pathogenesis. While the disease is likely multi-factorial, evidence is accumulating to implicate viruses as co-factors (either as initiating or exacerbating agents) of fibrotic lung disease. This review summarizes the available clinical and experimental observations that form the basis for the hypothesis that viral infections may augment fibrotic responses. We review the data suggesting a link between hepatitis C virus, adenovirus, human cytomegalovirus and, in particular, the Epstein-Barr gammaherpesvirus, in IPF. In addition, we highlight the recent associations made between gammaherpesvirus infection and lung fibrosis in horses and discuss the various murine models that have been used to investigate the contribution of gammaherpesviruses to fibrotic progression. We review the work demonstrating that gammaherpesvirus infection of Th2-biased mice leads to multi-organ fibrosis and highlight studies showing that gammaherpesviral infections of mice either pre- or post-fibrotic challenge can augment the development of fibrosis. Finally, we discuss potential mechanisms whereby viral infections may amplify the development of fibrosis. While none of these studies prove causality, we believe the evidence suggests that viral infections should be considered as potential initiators or exacerbating agents in at least some cases of IPF and thereby justify further study

Estimates for critical groups of solutions to quasilinear elliptic systems.

In this work we study a class of functionals, defined on Banach spaces, associated with quasilinear elliptic systems. Firstly, we prove some regularity results about the critical points of such functionals and then we estimate the critical groups in each critical point via its Morse index

Rapporti di lavoro non standard: impiego, utilità e criticità

Il presente articolo si pone l’obiettivo di effettuare un’analisi quantitativa e strutturale relativa alle diverse tipologie di lavoro non standard previste dalla normativa vigente nel panorama giuslavoristico italiano, evidenzindone in modo netto ed inequivocabile non solo gli aspetti caratteristici, al punto da riflettere indirettamente alcune peculiarità del mercato del lavoro italiano, ma anche la capacità di soddisfare in parte la domanda di lavoro da parte delle fasce più deboli della popolazione. Per tale analisi sono stati utilizzati gli ultimi dati disponibili prodotti dalla Rilevazione Continua delle Forze di Lavoro

CRITICAL GROUP ESTIMATES FOR NONREGULAR CRITICAL POINTS OF FUNCTIONALS ASSOCIATED WITH QUASILINEAR ELLIPTIC EQUATIONS

Abstract. We consider a class of quasilinear elliptic equations whose principal part includes the p-area and the p-Laplace operators, when p lies in a suitable left neighborhood of 2. For the critical points of the associated functional, we provide estimates of the corresponding critical groups, under assumptions that do not guarantee any further regularity of the critical point