Bounded solutions of the stationary Schrödinger equation with finite energy integral on model manifolds

Conditions for the existence of nontrivial bounded solutions of the stationary Schrodinger equation with a finite energy integral on model varieties are obtained. A condition for the existence of nontrivial bounded solutions with a finite integral of energy in the exterior of a compactum on arbitrary Riemannian manifolds is also obtained. Let $D=(0;+\infty)\times S,$ where $S$ is compact Riemannian manifold. Metrics on $D$ is following
$$ds^2=dr^2+g^2(r)d\theta^2.$$
Where $g(r)$ is positive, smooth on $(0,+\infty)$ function, $d\theta^2$ is metrics on $S.$ We will study solutions of the stationary Schrodinger equation
$$\Delta u-c(r)u=0$$
on $D$. Let $r_0=\mathrm{const} >0, n=\dim D$.