On Solvability in the Sense of Sequences for some Non-Fredholm Operators in Higher Dimensions
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Journal of Mathematical Sciences, Vol. 247, No. 6, June, 2020
ON SOLVABILITY IN THE SENSE OF SEQUENCES FOR SOME NON-FREDHOLM OPERATORS IN HIGHER DIMENSIONS V. Vougalter University of Toronto 27 King’s College Circle Toronto, Ontario M5S 1A1, Canada [email protected]
UDC 517.9
We study the solvability of some linear inhomogeneous elliptic equations and establish that, under reasonable technical conditions, the convergence in L2 (Rd ) of their righthand sides yields the existence and convergence of the solutions in L2 (Rd ). Bibliography: 29 titles.
1
Introduction
We consider the equation −Δu + V (x)u − au = f,
(1.1)
where u ∈ E = H 2 (Rd ), f ∈ F = L2 (Rd ), d ∈ N, a is a constant, and the scalar potential function V (x) tends to 0 at infinity. For a 0 the essential spectrum of the operator A : E → F corresponding to the left-hand side of Equation (1.1) contains the origin. As a consequence, such an operator does not satisfy the Fredholm property. Its image is not closed, for d > 1 the dimension of the kernel and the codimension of the image of A are not finite. This paper is devoted to the study of some properties of such operators. We recall that elliptic problems with non-Fredholm operators were extensively treated in recent years (cf., for example, [1]–[11]) along with potential applications to the theory of reaction-diffusion equations (cf., for example, [12, 13]). Non-Fredholm operators are also crucial in the study of wave systems with an infinite number of localized traveling waves (cf. [14]). In particular, in the case a = 0, the operator A satisfies the Fredholm property in some properly chosen weighted spaces (cf. [15]–[18] and [11]). However, the case a = 0 is significantly different and the method developed in these articles cannot be used. One of the important issues about equations with non-Fredholm operators concerns their solvability. Let us address it in the following setting. Let fn be a sequence of functions in the image of the operator A such that fn → f in L2 (Rd ) as n → ∞. Denote by un a sequence of functions from H 2 (Rd ) such that Aun = fn , n ∈ N. Since the operator A fails to satisfy the Fredholm property, the sequence un cannot be convergent. A sequence un such that Aun → f Translated from Problemy Matematicheskogo Analiza 102, 2020, pp. 85-96. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2476-0850
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is called a solution in the sense of sequences of the equation Au = f (cf. [1]). If such a sequence converges to a function u0 in the E-norm, then u0 is a solution to this problem. In this case, the solution in the sense of sequences is equivalent to the solution in the usual sense. However, in the case of non-Fredholm operators, the convergence cannot hold or can occur in some weaker sense. In this case, the solution in the sense of sequences cannot imply the existence of the usual solution. In the present paper, we find sufficient conditions of the equivalence of solutions in the sense of sequences and the usual solutions. In other words, we find conditions on seq
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