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{\bf A b s t r a c t.} { Positive solutions of fourth-order quasilinear differentialequation$$\left(p(t)|x''(t)|^{\alpha-1}\,x''(t)\right)^{\prime\prime}+q(t)|x(t)|^{\beta-1}\,x(t)=0,\quad \alpha>\beta>0,\leqno{(E)}$$is studied under the assumption that functions $p(t)$ and $q(t)$ are positive, continuous functions satisfying conditions $$ \int_a^\infty \frac{t}{p(t)^\frac{1}{\alpha}}\;dt=\infty,\:\:\:        \int_a^\infty \left(\frac{t}{p(t)}\right)^\frac{1}{\alpha}\,dt=\infty\,.   $$The main objective is to discuss the existence and precise asymptotic behavior of intermediate solutions of (E). First,  necessary and sufficient conditions for the existence of such solutions will be presented, and afterwards  focusing attention to equation (E) with {\it generalized  regularly varying coefficients}  $p(t)$, $q(t)$ and to its {\it generalized  regularly varying solution}, a complete information about  the structure and the asymptotic behavior of positive solutions will be given, using Karamata's theory of regular variation.

Emden-Fowler differential equations, regular varying functions, positive solutions