Combined Effects in Singular Elliptic Problems in Punctured Domain

The paper deals with nonlinear elliptic differential equations subject to some boundary value conditions in a regular bounded punctured domain. By means of properties of slowly regularly varying functions at zero and the Schauder fixed-point theorem, we establish the existence of a positive continuous solution for the suggested problem. Global estimates on such solution, which could blow up at the origin, are also obtained.

Even Order Half-Linear Differential Equations with Regularly Varying Coefficients

We establish nonoscillation criterion for the even order half-linear differential equation (−1)nfn(t)Φx(n)(n)+∑l=1n(−1)n−lβn−lfn−l(t)Φx(n−l)(n−l)=0, where β0,β1,…,βn−1 are real numbers, n∈N, Φ(s)=sp−1sgns for s∈R, p∈(1,∞) and fn−l is a regularly varying (at infinity) function of the index α−lp for l=0,1,…,n and α∈R. This equation can be understood as a generalization of the even order Euler type half-linear differential equation. We obtain this Euler type equation by rewriting the equation above as follows: the terms fn(t) and fn−l(t) are replaced by the tα and tα−lp, respectively. Unlike in other texts dealing with the Euler type equation, in this article an approach based on the theory of regularly varying functions is used. We establish a nonoscillation criterion by utilizing the variational technique.

On a limit structure of the Galton–Watson branching processes with regularly varying generating functions

We investigate limit properties of discrete time branching processes with application of the theory of regularly varying functions in the sense of Karamata. In the critical situation we suppose that the offspring probability generating function has an infinite second moment but its tail regularly varies. In the noncritical case, the finite moment of type Ε[x ln x] is required. The lemma on the asymptotic representation of the generating function of the process and its differential analogue will underlie our conclusions.

Existence and asymptotic behavior of intermediate type of positive solutions of fourth-order nonlinear differential equations

Under the assumptions that p and q are regularly varying functions satisfying conditions ??a t/p(t)1/? dt < ? and ??a (t/p(t))1/? dt = ? existence and asymptotic form of regularly varying intermediate solutions are studied for a fourth-order quasilinear differential equation (p(t)jx??(t)|?-1 x??(t))?? + q(t)|x(t)|?-1 x(t) = 0, ? > ? > 0. It is shown that under certain integral conditions there exist two types of intermediate solutions which according to their asymptotic behavior is to be divided into six mutual distinctive classes, while asymptotic behavior of each member of any of these classes is governed by a unique explicit law.