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

We consider the singular boundary value problem -r(x)y′(x) + q(x)y(x) = f(x), x ε R lim_|x|→∞ y(x) = 0, where f ∈ L_p(ℝ), p ∈ [1, ∞] (L_∞(ℝ) := C(ℝ)), r is a continuous positive function on ℝ, 0 ≤ q ∈ L₁^loc. 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 minimal requirements for r and q under which the above problem is correctly solvable in L_p(ℝ).