Hartman-Wintner type theorem for PDE with p-Laplacian

The well known Hartman-Wintner oscillation criterion is extended to the PDE $$div(||\nabla u||^{p-2}\nabla u)+c(x)|u|^{p-2}u=0\quad p>1\tag{E}$$ The condition on the function $c(x)$ under which (E) has no solution positive for large $||x||$, i.e. $\infty$ belongs to the closure of the set of zeros of every solution defined on the domain $\Omega=\{x\in R^n: ||x||>1\}$, is derived

On Constants in Nonoscillation Criteria for Half-Linear Differential Equations

We study the half-linear differential equation (r(t)Φ(x′))′+c(t)Φ(x)=0, where Φ(x)=|x|p−2x, p>1. Using the modified Riccati technique, we derive new nonoscillation criteria for this equation. The results are closely related to the classical Hille-Nehari criteria and allow to replace the fixed constants in known nonoscillation criteria by a certain one-parametric expression

Oscillation of Half-Linear Differential Equations with Delay

We study the half-linear delay differential equation , , We establish a new a priori bound for the nonoscillatory solution of this equation and utilize this bound to derive new oscillation criteria for this equation in terms of oscillation criteria for an ordinary half-linear differential equation. The presented results extend and improve previous results of other authors. An extension to neutral equations is also provided

On Eventually Positive Solutions of Quasilinear Second-Order Neutral Differential Equations

We study the second-order neutral delay differential equation [r(t)Φγ(z′(t))]′+q(t)Φβ(x(σ(t)))=0, where Φα(t)=|t|α-1t, α≥1 and z(t)=x(t)+p(t)x(τ(t)). Based on the conversion into a certain first-order delay differential equation we provide sufficient conditions for nonexistence of eventually positive solutions of two different types. We cover both cases of convergent and divergent integral ∫∞r-1/γ(t)dt. A suitable combination of our results yields new oscillation criteria for this equation. Examples are shown to exhibit that our results improve related results published recently by several authors. The results are new even in the linear case

Positive solutions of inequality with pp-Laplacian in exterior domains

summary:In the paper the differential inequality $$\Delta _p u+B(x,u)\le 0,$$ where $\Delta _p u=\div (\Vert \nabla u\Vert ^{p-2}\nabla u)$, $p>1$, $B(x,u)\in C(\mathbb{R}^{n}\times \mathbb{R},\mathbb{R})$ is studied. Sufficient conditions on the function $B(x,u)$ are established, which guarantee nonexistence of an eventually positive solution. The generalized Riccati transformation is the main tool

Discrete singular functionals

summary:In the paper the discrete version of the Morse’s singularity condition is established. This condition ensures that the discrete functional over the unbounded interval is positive semidefinite on the class of the admissible functions. Two types of admissibility are considered