Necessary and sufficient conditions for the solvability of a singular Dirichlet boundary problem for the Sturm-Liouville equation of general form

We consider the boundary problem(1)$-{\left(r\right(x\left)y^{\prime }\right(x\left)\right)}^{\prime }+q\left(x\right)y\left(x\right)=f\left(x\right),x\in R,$(2)$\underset{\left|x\right|\to \infty }{\lim }y\left(x\right)=0$ under the following conditions:

• 1)
  $r>0,\frac{1}{r}\in L_1^{\mathrm{loc}}\left(R\right),q\in L_1^{\mathrm{loc}}\left(R\right)$;
• 2)
  equation (1) is correctly solvable in $L_p\left(R\right)$, $p\in (1,\infty )$.

We obtain necessary and sufficient requirements for the functions r and q under which, regardless of the choice of a function $f\in L_p\left(R\right)$, $p\in (1,\infty )$, the solution $y\in L_p\left(R\right)$ of equation (1) satisfies (2).