Conditions for correct solvability of a simplest singular boundary value problem of general form. II (Q2475492)

Summary: We consider the singular boundary value problem \[ -r(x)y'(x) + q(x)y(x) = f(x),\quad x \in \mathbb R,\tag{1} \] \[ \lim_{|x|\to\infty} y(x)= 0,\tag{2} \] where \(f\in L_p(\mathbb R)\), \(p\in [1,\infty]\) \((L_{\infty}(\mathbb R) := C(\mathbb R))\), \(r\) is a continuous positive function on \(\mathbb R\), \(0\leq q\leq L^{\text{loc}}_1(\mathbb R)\). 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 \(L_p(\mathbb R)\) if for any function \(f\in L_p(\mathbb R)\) it has a unique solution \(y\in L_p(\mathbb R)\) and if the following inequality holds with an absolute constant \(c_p\in (0,\infty)\): \[ \|y\|_{L_p(\mathbb R)}\leq c_p\|f\|_{L_p(\mathbb R)},\quad \forall f\in L_p(\mathbb R). \] We find a relationship between \(r\), \(q\), and the parameter \(p\in [1,\infty]\), which guarantees the correctly solvability of the problem (1) and (2) in \(L_p(\mathbb R)\). For Part I, see ibid. 25, No. 2, 205--235 (2006; Zbl 1122.34021).