Conditions for correct solvability of a simplest singular boundary value problem of general form. II

We consider the singular boundary value problem -r(x)y′(x) + q(x)y(x) = f(x), x ∈ ℝ (1) lim y(x) = 0, (2) |x|→∞ where f ∈ L_p(ℝ), p ∈ [1, ∞] (L_∞(ℝ):= C(ℝ)), r is a continuous positive function on ℝ, 0 ≤ q ∈ L₁^loc (ℝ). A solution of this problem is, by definition, any absolutely continuous function y satisfying the limit condition and almost everywhere the differential equation. This problem is called correctly solvable in a given space L_p(ℝ) if for any function f ∈ L_p(ℝ) it has a unique solution y ∈ L _p(ℝ) and if the following inequality holds with an absolute constant c_p ∈ (0, ∞): ||y||L_p(ℝ) ≤ c_p||f||L_p(ℝ), ∀ f L_p(ℝ). We find a relationship between r, q, and the parameter p ∈ [1, ∞], which guarantees the correctly solvability of the problem (1) and (2) in L _p(ℝ).