### Hartman-Wintner type theorem for PDE with $p$-Laplacian.

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In this paper we study extremal properties of functional associated with the half–linear second order differential equation E${}_{p}$. Necessary and sufficient condition for nonnegativity of this functional is given in two special cases: the first case is when both points are regular and the second is the case, when one end point is singular. The obtained results extend the theory of quadratic functionals.

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.

In the paper the differential inequality $${\Delta}_{p}u+B(x,u)\le 0,$$ where ${\Delta}_{p}u={div(\parallel \nabla u\parallel}^{p-2}\nabla u)$, $p>1$, $B(x,u)\in C({\mathbb{R}}^{n}\times \mathbb{R},\mathbb{R})$ 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.

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