Nonzero bounded solutions of one class of~nonlinear ordinary differential equations

È. M. Muhamadiev ; A. N. Naimov

Sbornik. Mathematics, Tome 202 (2011) no. 9, pp. 1373-1386

Résumé

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 $c>-1/4$, $k>1$, $02\alpha$, 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. Bibliography: 7 titles.

Détail

Keywords: differential equation, function space, weakly continuous functional, eigenfunction of the gradient of a functional.