Topological degree for a class of elliptic operators in ${\mathbb R}^n$

Abstract

A class of elliptic operators in R n is considered. It is proved that the operators are Fredholm and proper. The topological degree is constructed. Existence of solutions for a reaction-diffusion system is studied.

Fredholm property of general elliptic problems

“…Limiting operators were first considered in [15], [20], [21] for differential operators on the real axis with quasi-periodic coefficients, and then for elliptic operators in R n or for domains cylindrical or conical at infinity [8], [25], [26], [39].…”

“…Limiting operators and their interrelation with solvability conditions and with the Fredholm property were first studied in [15], [20], [21] (see also [39]) for differential operators on the real axis, and later for some classes of elliptic operators in R n [8], [25], [26], in cylindrical domains [9], [43], or in some specially constructed domains [6], [7]. Some of these results are obtained for the scalar case, some others for the vector case, under the assumption that the coefficients of the operator stabilize at infinity or without this assumption.…”

Properness and topological degree for general elliptic operators

“…To formulate it we define limiting problems. In the simplest case where Ω = R 1 , has a nonzero solution, belongs to the essential spectrum of the operator L. Limiting problems and the normal solvability for linear elliptic operators in unbounded domains were studied in a number of works for R n , and for domains with cylindrical and conical ends (see [3,21,25,26,33] and the references therein).…”

Solvability conditions for elliptic problems with non-Fredholm operators

“…The second convergence follows from the assumptions on the sequence {(x (2) i , y (2) i )} and from the positiveness of v(y). On the other hand, multiplying the equality…”

“…To determine it explicitly, we make some simplifying assumptions. We assume the existence of the limits a(x, y) → a If one of them has a nonzero solution in C 2+α (R 2 ), then the corresponding value of λ belongs to the essential spectrum of the operator L, i.e., the operator L − λI is not Fredholm [6], [2].…”