This paper is devoted to studies of non-negative, non-trivial (classical, punctured, or distributional) solutions the higher order Hardy-H\'enon equations \[ (-\Delta)^m u = |x|^\sigma u^p \] in $\mathbf R^n$ with $p > 1$. We show that condition n - 2m \frac{2m+\sigma}{p-1} >0 necessary for existence distributional solutions. For $n \geq 2m$ and $\sigma -2m$, we prove any solution satisfies an ...

For the connections model of strategic network formation, with two-way flow of information and without information decay, specific parameter configurations are given for which Nash networks do not exist. Moreover, existence and the scope of Nash network architectures are briefly discussed. JEL Classification Numbers C72 · D85

We consider polynomial spline spaces S r d (4) of degree d and smoothness r deened on triangulations. It is known that for d 3r + 2, S r d (4) possesses a basis of star-supported splines, i.e., splines whose supports are at most the set of triangles surrounding a vertex. Here we extend the theory by showing that for all d 3r + 1, there exist triangulations for which no such bases exist. x1. Int...

In this article we prove a collection of new non-linear and non-local integral inequalities. As an example for u ≥ 0 and p ∈ (0,∞) we obtain ∫ R3 dx u(x) ≤ ( p+ 1 p )2 ∫ R3 dx {(−4)−1u(x)}|∇u p 2 (x)|. We use these inequalities to deduce global existence of solutions to a non-local heat equation with a quadratic non-linearity for large radial monotonic positive initial conditions. Specifically,...

the existence is a categorical concept according to kant’s philosophy. kant, going to be a subjectivist philosopher, negates the existence as a perfection and concludes the separation between the mind and extra-mental world. but the externality is not an origin which we deny it easily, even if it is faced with multiple problems such as the issue of simplicity and multiplicity and also being his...