Nonzero bounded solutions of one class of nonlinear ordinary differential equations

The paper is concerned with an ordinary differential equation of the form -ψ''(x)+(1+c/x2)ψ(x)=1/xα|ψ(x)|k-1ψ(x), x>0; (1) where k and α are positive parameters, k>1, and c is a constant, subject to the boundary condition ψ(0)=0, ψ(+∞)=0; (2) A variational approach based on finding the eigenvalues of the gradient of the functional Fk,α(f)=∫0+∞|f(s)|k+1s-α ds acting on the space of absolutely continuous functions H01={f:f,f' element of L2(0,+∞), f(0)=0} is used to show that if c>-1/4, k>1, 0<2α<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→0 and x→+∞ are obtained. Bibliography: 7 titles.