We investigate the uniqueness of nonnegative solutions to the following differential inequality div(A(x)|∇u|∇u) + V (x)u1 |∇u|2 ≤ 0, (1) on a noncompact complete Riemannian manifold, where A, V are positive measurable functions, m > 1, and σ1, σ2 ≥ 0 are parameters such that σ1 + σ2 > m− 1. Our purpose is to establish the uniqueness of nonnegative solution to (1) via very natural geometric assu...

In this paper, the existence and nonexistence of nonnegative entire large solutions for the quasilinear elliptic equation   2 | | = ( ) ( ) ( ) ( ) m div u u p x f u q x g u     are established, where 2 m  , f and g are nondecreasing and vanish at the origin. The locally H older continuous functions p and q are nonnegative. We extend results previously obtained for special cases of f a...

We prove the existence of positive solutions to the nonlinear parabolic equation ∆u− ∂u ∂t = p(x, t)f(u) in the half space R+, n ≥ 2, subject to Dirichlet boundary conditions. The function f is nonnegative continuous non-increasing, and the potential p is nonnegative and satisfies some hypotheses related to the parabolic Kato class. We use potential theory arguments to prove our main result.

In Nonnegative Matrix Factorization (NMF), a nonnegative matrix is approximated by a product of lower-rank factorizing matrices. Most NMF methods assume that each factorizing matrix appears only once in the approximation, thus the approximation is linear in the factorizing matrices. We present a new class of approximative NMF methods, called Quadratic Nonnegative Matrix Factorization (QNMF), wh...

We study a free boundary problem describing the propagation of laminar flames. The problem arises as the limit of a singular perturbation problem. We introduce the notion of viscosity solutions for the problem to show the maximum principle-type property of the solutions. Using this property we show the uniform convergence of the approximating solutions and the uniqueness of the viscosity soluti...

where Ω is a bounded domain in R , N ≥ 3, with a smooth boundary ∂Ω and f : Ω× R× R → R. The existence of positive solutions to (1.1) in the case where f depends only on u and grows subcritically has been studied extensively in recent years (see the review article by Lions [3] and the references therein). In this paper, we establish the existence of nonnegative solutions to (1.1) where f has a ...

The paper deals with the existence and nonexistence of nontrivial nonnegative solutions for the sublinear quasilinear system div (|∇ui |p−2∇ui)+ λfi(u1, . . . , un)= 0 in Ω, ui = 0 on ∂Ω, i = 1, . . . , n, where p > 1, Ω is a bounded domain in RN (N 2) with smooth boundary, and fi , i = 1, . . . , n, are continuous, nonnegative functions. Let u = (u1, . . . , un), ‖u‖ = ∑n i=1 |ui |, we prove t...

This note provides a simple sufficient condition ensuring that solutions of stochastic delay differential equations (SDDEs) driven by subordinators are nonnegative. While, to the best our knowledge, no nonnegativity conditions available in context SDDEs, we compare result literature within subclass invertible continuous-time ARMA (CARMA) processes. In particular, analyze why cannot be necessary...