A population biological model with a singular nonlinearity

summary:We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form $$\begin {cases} -{\rm div}(|x|^{-\alpha p}|\nabla u|^{p-2}\nabla u)=|x|^{-(\alpha +1)p+\beta } \Big (a u^{p-1}-f(u)-\dfrac {c}{u^{\gamma }}\Big ), \quad x\in \Omega ,\\ u=0, \quad x\in \partial \Omega , \end {cases}$$ where $\Omega$ is a bounded smooth domain of ${\mathbb R}^N$ with $0\in \Omega$, $1<p<N$, $0\leq \alpha < {(N-p)}/{p}$, $\gamma \in (0,1)$, and $a$, $\beta$, $c$ and $\lambda$ are positive parameters. Here $f\colon [0,\infty )\to {\mathbb R}$ is a continuous function. This model arises in the studies of population biology of one species with $u$ representing the concentration of the species. We discuss the existence of a positive solution when $f$ satisfies certain additional conditions. We use the method of sub-supersolutions to establish our results

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

Proofs of Urysohn's lemma and the Tietze extension theorem via the Cantor function

Urysohn's Lemma is a crucial property of normal spaces that deals with separation of closed sets by continuous functions. It is also a fundamental ingredient in proving the Tietze Extension Theorem, another property of normal spaces that deals with the existence of extensions of continuous functions. Using the Cantor function, we give alternative proofs for Urysohn's Lemma and the Tietze Extension Theorem

Modular Construction of Modal Logics

We present a modular approach to defining logics for a wide variety of state-based systems. We use coalgebras to model the behaviour of systems, and modal logics to specify behavioural properties of systems. We show that the syntax, semantics and proof systems associated to such logics can all be derived in a modular way. Moreover, we show that the logics thus obtained inherit soundness, completeness and expressiveness properties from their building blocks. We apply these techniques to derive sound, complete and expressive logics for a wide variety of probabilistic systems

CD1-010

Swadesh Indo-Aryan word listNagar Valley, Gilgit-Baltistan Province, Pakista