Nonzero bounded solutions of one class of nonlinear ordinary differential equations

The paper is concerned with an ordinary differential equation of the form
$$\begin{equation} -\psi''(x)+\biggl(1+\frac c{x^2}\biggr)\psi(x)= \frac1{x^\alpha}|\psi(x)|^{k-1}\psi(x), \qquad x>0, \tag{1} \end{equation}$$
where $k$ and $\alpha$ are positive parameters, $k>1$, and $c$ is a constant, subject to the boundary condition
$$\begin{equation} \psi(0)=0, \qquad \psi(+\infty)=0. \tag{2} \end{equation}$$
A variational approach based on finding the eigenvalues of the gradient of the functional $F_{k,\alpha}(f)=\displaystyle\int_0^{+\infty}|f(s)|^{k+1}s^{-\alpha}\,ds$ acting on the space of absolutely continuous functions $H_0^1=\{f:f,f'\in L_2(0,+\infty), f(0)=0\}$ is used to show that if $c>-1/4$, $k>1$, $0<2\alpha<k+3$, then problem $(1)$, $(2)$ has a countable number of nonzero solutions, at least one of which is positive. For nonzero solutions, asymptotic formulae as $x\to0$ and $x\to+\infty$ are obtained.
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