Existence of blow-up solutions for a non-linear equation with gradient term in RN

AbstractIn this paper we study the existence of positive large solutions for the equation Δpu+λ|∇u|p−1=ρ(x)f(u) in RN, where f is a non-negative non-decreasing function and ρ is a non-negative continuous function. We show under some hypotheses detailed below the existence of positive solutions which blow up at infinity

A new critical curve for the Lane-Emden system

We study stable positive radially symmetric solutions for the Lane-Emden system $-\Delta u=v^p$ in $\R^N$, $-\Delta v=u^q$ in $\R^N$, where $p,q\geq 1$. We obtain a new critical curve that optimally describes the existence of such solutions.Comment: 13 pages, 1 figur

Dark clouds in co-creation, and their silver linings practical challenges we faced in a participatory project in a resource-constrained community in India, and how we overcame (some of) them

BACKGROUND: While any type of field-based research is challenging, building action-oriented, participatory research in resource-constrained settings can be even more so. OBJECTIVE: In this article, we aim to examine and provide insights into some of the practical challenges that were faced during the course of a participatory project based in two non-notified slums in Bangalore, India, aiming to build solutions to indoor air pollution from cooking on traditional cook stoves. METHODS: The article draws upon experiences of the authors as field researchers engaged in a community-based project that adopted an exploratory, iterative design to its planning and implementation, which involved community visits, semi-structured interviews, prioritization workshops, community forums, photo voice activities, chulha-building sessions and cooking trials. RESULTS: The main obstacles to field work were linked to fostering open, continued dialogue with the community, aimed at bridging the gap between the 'scientific' and the 'local' worlds. Language and cultural barriers led to a reliance on interpreters, which affected both the quality of the interaction as well as the relationship between the researchers and the community that was built out of that interaction. The transience in housing and location of members of the community also led to difficulties in following up on incomplete information. Furthermore, facilitating meaningful participation from the people within the context of restricted resources, differing priorities, and socio-cultural diversity was particularly challenging. These were further compounded by the constraints of time and finances brought on by the embeddedness of the project within institutional frameworks and conventional research requirements of a fixed, pre-planned and externally determined focus, timeline, activities and benchmarks for the project. CONCLUSIONS: This article calls for revisiting of scientific conventions and funding prerequisites, in order to create spaces that support flexible, emergent and adaptive field-based research projects which can respond effectively to the needs and priorities of the community

Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient

In this paper, we study a nonlinear Dirichlet problem of p-Laplacian type with combined effects of nonlinear singular and convection terms. An existence theorem for positive solutions is established as well as the compactness of solution set. Our approach is based on Leray-Schauder alternative principle, method of sub-supersolution, nonlinear regularity, truncation techniques, and set-valued analysis

An electron microscopy study of graphite growth in nodular cast irons

Growth of graphite during solidification and high-temperature solid-state transformation has been investigated in samples cut out from a thin-wall casting which solidified partly in the stable (iron–graphite) and partly in the metastable (iron–cementite) systems. Transmission electron microscopy has been used to characterize graphite nodules in as-cast state and in samples having been fully graphitized at various temperatures in the austenite field. Nodules in the as-cast material show a twofold structure characterized by an inner zone where graphite is disoriented and an outer zone where it is well crystallized. In heat-treated samples, graphite nodules consist of well-crystallized sectors radiating from the nucleus. These observations suggest that the disoriented zone appears because of mechanical deformation when the liquid contracts during its solidification in the metastable system. During heat-treatment, the graphite in this zone recrystallizes. In turn, it can be concluded that nodular graphite growth mechanism is the same during solidification and solid-state transformatio

A guide to the Choquard equation

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$-\Delta u + V(x)u = \bigl(|x|^{-(N-\alpha)} * |u|^p\bigr)|u|^{p - 2} u \qquad \text{in $\mathbb{R}^N$},$$ and some of its variants and extensions.Comment: 39 page