Decreasing Regularly Varying Solutions of Sublinearly Perturbed Superlinear Thomas–Fermi Equation

Abstract

We consider the perturbed Thomas–Fermi equation

$$\begin{array}{ll} x^{\prime \prime}\, =\, p(t)|x|^{\gamma-1}x\, +\, q(t)|x|^{\delta-1}x, \qquad \qquad \qquad (A) \end{array}$$

where δ and γ are positive constants with ${\delta < 1 < \gamma}$ and p(t) and q(t) are positive continuous functions on ${[a,\infty), a > 0}$ . Our aim here is to establish criteria for the existence of positive solutions of (A) decreasing to zero as ${t \to \infty}$ in the case where p(t) and q(t) are regularly varying functions (in the sense of Karamata). Generalization of the obtained results to equations of the form

$$\begin{array}{ll} \left(r(t)x^{\prime}\right)^{\prime}\, =\, p(t)|x|^{\gamma-1}x \,+ \,q(t)|x|^{\delta-1}x, \qquad \qquad \qquad (B) \end{array}$$

is also given.