Clear[ve]
r0=1;
ve[r_]:=-v0 Exp[-(r/r0)];
Plot[ve[r],{r,0,5}]

e=1;
eqn=NDSolve[{u''[r]+(e-ve[r])u[r]==0,u[0]==0,u'[0]==1},u[r],{r,-3,3}]
Plot[u[r]/.eqn,{r,0,5}]

e=3;
eqn=NDSolve[{u''[r]+(e-vs[r])u[r]==0,u[0]==0,u'[0]==1},u[r],{r,-3,3}]
Plot[u[r]/.eqn,{r,0,5}]