Fibering Method Research Articles

On asymptotics of solutions to semilinear elliptic equations near the first eigenvalue of the nonperturbed problem

The following elliptic equations withp-Laplacian $$ - \Delta _p u = \lambda g(x)\left| u \right|^{p - 2} u + f(x)\left| u \right|^{\gamma - 2} u$$ are considered in the entire space ℝ N and in the bounded domain with the Dirichlet boundary conditions. By the fibering method for the basic positive solutions of these equations, we derive the following asymptotic formula $$u^\lambda = (\lambda _1 - \lambda )^{1/(\gamma - p)} u_1 + o((\lambda _1 - \lambda )^{1/(\gamma - p)} )$$ for λ↑λ1, where λ1 is the first eigenvalue and u1 is the corresponding eigenfunction of nonperturbed problem (ƒ=0).

ON THE EXISTENCE OF A COUNTABLE SET OF SOLUTIONS IN A PROBLEM OF POLARON THEORY

An investigation is made of a certain nonlinear second-order integro-differential equation having numerous applications in various areas of physics (polaron theory, the theory of many-particle quantum systems, and so on). Under certain assumptions it is proved that there is a positive solution, and, moreover, an infinite set of distinct solutions. Use is made of the Lyusternik-Shnirel'man theory of critical points and the fibering method of S. I. Pokhozhaev.

Fibering method in some probabilistic problems

The paper is devoted to a systematic study of the distributions of functionals of stochastic processes by the fibering method and to a survey of results obtained in this direction in recent years. Principal attention is given to distinguishing conditions ensuring: a) absolute continuity; b) the existence of a bounded density; c) applicability of the local limit theorem for the distributions of functionals. Smooth, convex functionals and functionals of integral type are considered in detail.